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I will discuss some aspects of the recent proof that there are arbitrarily long arithmetic progressions of primes, which is joint work with Terry Tao. I will also discuss some more recent work of ours, which gives an asymptotic for the number of 4-term APs of primes, all less than N.
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We start with some examples of singularities that algebraic varieties can have in low dimensions. We look at resolutions of singularities, that is, mappings from a smooth manifold onto a singular space. One can try to define invariants of a singular space using known invariants of its resolution. I will describe some successful invariants of this type: intersection homology theory, the elliptic genus, and stringy Betti numbers.
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The talk aims to explain the connection between the following topics: - Canonical Bases for Kac-Moody Lie Algebras - Representation Theory of Preprojective Algebras - Varieties of Modules.
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This will mainly be about SL(2,C) and SL(2,R). The first point of view is that of Harish-Chandra, and leads to irreducible unitary representations, the Plancherel measure and the tempered dual. The second point of view is that of Connes and Kasparov, which leads from the reduced C*-algebra back to the representations rings R(SU(2)) and R(SO(2)). I will relate these two points of view, and describe recent results for GL(3) (joint work with Anne-Marie Aubert).
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In this talk I will give a brief introduction to the well known Catalan number and present research results associated with them, some of which are set in historical context. Two recently announced Catalan-related sequences which arise from elliptic integrals---namely, the so called Catalan-Larcombe-French and Fennessey-Larcombe-French---are then introduced, and their properties discussed.
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In the talk I will show that one may predict the character of instabilities of robot's behavior knowing the cohomology algebra of its configuration space.
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We discuss relations between the K-theory of number fields and their zeta functions, both classically and (more conjecturally) p- adically. Apart from talking about the theoretical description of those regulators we also touch upon aspects on how to compute them in practice.
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It is well-known that a generalized cohomology theory applied to the infinite dimensional complex projective space $CP^\infty$ often gives rise to a one-dimensional group law. This fact has innumerable applications in stable homotopy theory. One particularly important class of a one dimensional formal group is associated with $K(n)^* CP^\infty)$ where $K(n)$ is the nth Morava K-theory. This is an essentially unique example of a one-dimensional formal group of height $n$ over a field. It turns out that if one replaces $CP^n=K(Z,2)$ with $K(Z,l) $, the integral Eilenberg-Mac Lane space with $\pi_l=Z$ then the corresponding object is a formal group of finite height (a.k.a. smooth p-divisible group). This is essentially a 25 year old result of Ravenel-Wilson although they did not phrase it in this way. Letting l vary we obtain a collection of p-divisible groups which possesses a remarkable symmetry. Particularly, any p- divisible group enters in this collection together with its Serre dual (an analogue of the notion of principal polarization for abelian schemes). This and related results are obtained by studying the Dieudonne modules associated to the corresponding p- divisible groups. This is a joint work of myself with Victor Buchstaber.
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Software testing is the most expensive and difficult part of the software production process. IBM, for example estimate that testing and reviewing activities account for at least 50% of any project, in safety critical projects it can reach 90%. The sales of software in the UK in 2001 was
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Let G be a finite group acting on an algebraic curve X. This action induces an action on various Riemann-Roch spaces such as the vector space of global holomorphic differentials on X. We determine these (modular) representations in local terms, thereby generalizing the classical Chevalley-Weil formula from characteristic 0 to the so-called weakly ramified case, an important case of wild ramification.
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While the classical Weil conjectures are concerned with counting points on varieties over finite fields, we consider the problem of counting points on \emph {difference} varieties over algebraic closures of finite fields with powers of Frobenius. This context is suitable e.g. for uniform treatment of Ree and Suzuki families of finite simple groups.
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Given a system of polynomial equations over a finite field, one may associate with it a finite dimensional vector space, known as the ``rigid cohomology'' of the system. This construction is very useful; for example, it allows one to prove good bounds on the number of solutions to the system over the finite field (Weil conjectures). The construction was first proposed in the 1960s; however, showing that the vector spaces it associated with systems were finite dimensional turned out to be very difficult. (This was not done until the mid 1990s, independently by Berthelot and Christol-Mebkhout.) In my talk I will discuss an ``effectivity problem'' related to the finiteness of the rigid cohomology of a system of equations.
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I'll discuss some joint work with Will Turner on certain algebras associated to tilings of the plane by rhombi. These `rhombal algebras' defined by Michael Peach were inspired by the representation theory of symmetric groups in positive characteristic. There is a close connection between the combinatorics of the tilings and the homological properties of the algebras. For example certain basic mutations of tilings correspond to equivalences of derived categories of modules.
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We define and study a Lusternik-Schnirelmann theory for orbifolds. The orbifold category provides a new invariant of the homotopy type of the orbifold that gives a numerical measure of the complexity of the orbifold $X$. In particular, the orbifold category gives a lower bound on the number of critical points of any orbifold smooth function $f\colon X\rightarrow R$. We use equivariant methods to find upper and lower bounds on the orbifold category in terms of the orbifold resolution of the singular set. We obtain a generalization of the classical cohomological lower bound for orbifold category using the orbifold cohomology theory constructed by Chen-Ruan.
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We consider a polynomial ring $k[x_1,...,x_n]$ over a finite field $k$ and suppose that some finite group $G$ acts on it by linear substitutions. We want to understand the ring as a $kG$-module. We present a structure theorem that describes this in a finite way. It has several notable corollaries, such as the fact that only finitely many indecomposable modules occur as summands (up to isomorphism) and the fact that we can write down an a priori bound on the degrees of the generators of the invariant subring.
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The first parts of my talk will be very elementary. I will introduce a deformation of the ordinary derivation of real functions of one variable. I will discuss the corresponding operator on polynomials (for which values of the deformation parameter are there non constant polynomials killed by the operator ?) and on analytic functions (spectrum of the operator, eigenfunctions as Bessel functions). Then, I will switch to the dimension $n$ case, where one has a commuting family of operators deforming the $d/dx_i$ (the Dunkl operators). I will focus on the action on polynomial functions of $n$ variables and explain how this is controlled by an algebra deforming the algebra of polynomial differential operators (a doubly degenerate double affine Hecke algbra). This leads to the study of representations of this algebra. I will describe how the representation theory of this algebra is studied, in analogy with the representation theory of the Lie algebra $gl_n(\mathbb{C})$. This last part brings a lot of exciting mathematics:
- Knizhnik-Zamolodchikov equations, braid groups, Hecke algebras to analyze representations as systems of differential equations.
- quantum general linear group, Fock spaces to describe precisely the representation theory
- Hilbert scheme of points on $\mathbb{C}^2$ as the geometric object connected to the representation theory.
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Let $F$ be a local field with finite residue field, ring of integers $O$, and maximal ideal $p$. Let $G$ be a reductive group scheme over $O$ (e.g. $G=GL_n$). We present an approach to the study of representations of the finite groups $G_{r}:=G(O/p^r)$, which for $r=1$ coincides with the theory of Deligne and Lusztig. One reason why such a study is of interest is the close connection between the representation theory of the groups $G_{r}$, and the representation theory of the group $G(F)$. One of the few cases where the representations of $G_{r}$ are known for all $r;eq1$, is when $G=GL_{2}$. This is due to several people, including Kutzko, and the method used is purely algebraic, and quite different from our geometric approach. We show how the two methods can be linked, and in particular how the algebraic method can be used to analyse representations constructed geometrically.
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A homolgy decomposition is a way to build a space out of 'simpler' space. A CW -complexes is given an iterated building process based on spheres and discs where as the gluing data for homology decompositions is encoded in a functor defined on a 'nice' category with values in the category of topological spaces, and where all simpler spaces are glued together in one step. Homology decompositions are one of the major tools to understand the homotopy theory of classifying spaces. We will apply these ideas in several much more algebraic contexts, Stanley-Reisner algebras associated to simplicial complexes, invariant theory and group cohomology.
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The $p$-local approach to finite group theory tries to understand the structure of a finite group in terms of one of its Sylow-$p$-subgroups $P$ (they are all isomorphic, so it doesn't matter which one we take) and the way in which $P$ is embedded into $G$. This approach goes well back to the early stages of the theory, illustrated by theorems of Burnside and Frobenius, and plays an important role in the context of the classification of finite simple groups. One can describe the p-local structure of $G$ in terms of a category, the fusion system of $G$. As a consequence of work of Alperin and Broue around 1980 it appears that categories with very similar formal properties occur also in modular representation theory, prompting Puig in the 1990's to formalise the notion of fusion systems independently of finite groups and Benson to speculate whether any such fusion system gives rise to a topological space which would play the role of classifying space of the group. Broto, Levi and Oliver developed in recent years the precise framework for topological spaces arising in this way - giving a sense to the concept of classifying spaces of finite groups which don't exist...
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According to Godel's theorem, formulated in 1930, no non- trivial theory of arithmetic can have its proof of consistency in terms of the presuppositions of the theory itself. This means that it is not possible to form a final form of mathematics that would be its sole form which is also necessarily true. Since physics has to be heavily mathematical, this also means the end of hopes that a final physical theory could ever be formulated. Contrary to a recent claim of Prof Hawking, this does not mean of the end of physics, though it constitutes a death blow at those hopes, often proposed with great arrogance. Godel's theorem is an assurance that the work of physicists will go on to no end.
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In topological quantum field theory one is interested in studying functors from a topological category of $n$-dimensional cobordisms into the category of vector spaces. In two dimensions such functors are very well understood. In fact, specifying a (symmetric monoidal) functor from the 2-dimensional cobordism category 2-Cob into Vect is equivalent to specifying a commutative Frobenius algebra. This makes the study of 2-dimensional TQFT's particularly simple. Recent developments in string theory have prompted many to consider topological quantum field theories using a more interesting version of the $2$-dimensional cobordism category, namely one that allows for cobordisms between $1$-manifolds with boundary. In this talk I will define a category of planar cobordisms between `open strings' and show that functors from this category into Vect are equivalent to (not necessarily commutative) Frobenius algebras. This result arises naturally by considering adjunctions in 2- categories. If time permits, I will also sketch how this process can be generalized to higher-dimensional surfaces using higher-dimensional category theory. This talk is intended to be accessible; all concepts from higher-dimensional category theory will be introduced in the talk.
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In algebraic topology one typically applies powerful algebraic invariants to encode homotopy properties of topological spaces. In certain cases it is possible and useful to reverse this process by assigning a space to an algebraic object. An instance of this is the assignment to a finite group $G$ of a classifying space $BG$, whence the group $G$ can be recovered as the fundamental group. Furthermore, group homomorphisms between finite groups correspond bijectively to homotopy classes of maps between their classifying spaces. In this talk I will discuss how this correspondence changes when we focus on properties relative to a prime $p$. Topologically this means applying the $p$-completion functor to $BG$. I will present three classification theorems for $p$-completed classifying spaces of finite groups. First, the unstable classification, predicted by Martino- Priddy and proved by Oliver, which classifies the homotopy type of the $p$-completed classifying space of $G$ via the fusion system of $G$. Second, the stable classification, due to Martino-Priddy, which classifies the stable homotopy type of the p-completed classifying space of $G$ via weaker data, which loosely speaking can be regarded as a linearisation of the fusion system. Finally, the partially stable classification, which links the unstable and stable classifications. This is the surprising result that, by keeping track of inclusions of Sylow subgroups, the stable homotopy type of the $p$-completed classifying space of $G$ can again be classified via the fusion data of $G$. This classification also gives a simple description of maps realising stable homotopy equivalences (while preserving the inclusions of Sylow subgroups).
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The hypothesis that the discrete substructure of spacetime is a causal set suggests a straightforward model building technique: invent phenomenological dynamics for matter (particles or fields) on a background causal set that is well approximated by our continuum spacetime. These models can be analysed to see if they predict observable eviations from continuum models. I will describe two examples of such models: "particle swerves" and a model of detector response to the scalar field of a scalar charge source.
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The signature of a manifold is an important invariant: it is the basic obstruction to a manifold being the boundary of a manifold of one dimension higher. The talk will survey some classical results for computing signatures and explain how, by introducing a notion of signature for certain singular spaces, we can obtain very geometric proofs and significant extensions of these results.
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The talk is devoted to associative N-graded algebras presented by n generators and n(n-1)/2 quadratic relations and satisfying the so-called Poincare-Birkhoff-Witt condition (PBW-algebras). We consider examples of such algebras depending on two continuous parameters (namely, on an elliptic curve and a point on this curve) which are flat deformations of the polynomial ring in n variables. Diverse properties of these algebras will be described, together with their relations to integrable systems, deformation quantization, moduli spaces and other directions of modern investigations.
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To a manifold M, string theory associates its topological partition function, a finite dimensional approximation to a complicated path integral on M. One of the simplest cases when this function can be computed explicitly is that of local P^1, a certain complex threefold fibred over the projective line. Its partition function can be written in six or seven different ways, as infinite sum or infinite product, related to Gromov-Witten theory, Donaldson-Thomas theory, the combinatorics of partitions, Chern-Simons theory of knots... The talk will introduce these ideas in elementary terms.
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I will discuss the role played by certain aspects of quantum field theory such as the Feynman calculus and the Batalin-Vilkovisky formalism in the construction of graph homology and cohomology classes, as introduced by Kontsevich in his 92/93 papers. I will also give the first example of a nontrivial pairing between a graph homology and cohomology class which arises from the evaluation of a super-integral, more than ten years since the idea was first proposed by Kontsevich.
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How can one construct computer networks without bottlenecks? Is there a method of efficiently shuffling a pack of cards? How does the spectrum of the Laplacian on a manifold behave under finite-sheeted covers? How can one detect `large' groups? Do hyperbolic 3-manifolds contain essential surfaces? In my talk, I will show how these questions are all related to an intriguing concept known as `Property tau'.
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Several decades ago the Betti numbers of the moduli spaces of stable vector bundles (with fixed mutually coprime rank and degree) over a Riemann surface were found, first by Harder and Narasimhan using number-theoretic methods and counting objects defined over finite fields, and soon after by Atiyah and Bott using Yang-Mills theory and equivariant Morse theory. This talk will link these two approaches and describe some more recent results on the geometry of the moduli spaces.
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Far beyond the realm where we can count "vertices minus edges", there are spaces that, nevertheless, appear to have a well- defined Euler characteristic. For example, the Julia set of any rational function f seems to have an Euler characteristic, a number giving basic information about the dynamical behaviour of f. But to define the Euler characteristic of such spaces, we first need to define the Euler characteristic of a category. This involves generalizing the Mobius inversion formula of classical number theory. We'll see, for instance, that the Euler characteristic of the category of finite sets and bijections is e = 2.718... . Throughout, our motto is: "Euler characteristic is generalized cardinality".
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The last few years have seens some spectacular developments in our understanding of hyperbolic 3-manifolds. The title of my talk is taken from a recent article in Science magazine on this topic. The problems are equivalent to much simpler sounding questions about what happens when you iterate Mobius maps. I will give an overview of the background and the significance of the new developments, illustrated with many pictures from our book Indra's Pearls (Mumford, Series and Wright, CUP 2002).
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I shall give an overview for non-experts of the modern theory of harmonic maps and how it applies to questions of classical (and sometimes unfashionable) differential geometry via an appropriate notion of Gauss map.
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In this talk we study combinatorial questions about primes. In particular, Ostmann asked whether there exist two sets A and B (with at least two elements each) so that their sumset A+B equals the set of primes, for sufficiently large primes. Using a new version of the large sieve method we show, that such sets A and B would need to have counting functions of size $N^{1/2 +o(1)}$, whereas previously only a lower bound of $N^{o(1)}$ and an upper bound of $N^{1+o(1)}$ was known. This implies, for example, that the set of primes cannot be decomposed into three such sets. This talk will give a nontechnical survey of the underlying ideas and show how a new type of the large sieve method and combinatorial counting arguments (including graph theory) can be applied to such problems. Other recent work on primes by Green, Tao, Goldston, Pintz and Yildirim will be mentioned
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The first part of the talk will be a short survey on possible approaches to and results on noncommutative deformations of algebraic verieties, in particular, projective spaces. In the second part we will discuss results and conjectures on the geometric structure of Poisson brackets on Fano varieties.
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Lawvere theories and monads have been the two main category theoretic formulations of universal algebra, Lawvere theories arising in 1963 and the connection with monads being established a few years later. Monads, although mathematically the less direct and less malleable formulation, rapidly gained precedence. A generation later, Eugenio Moggi instigated the use monads in theoretical computer science in order to model computational effects, without reference to universal algebra. But since then, the relevance of universal algebra to computational effects has been recognised, leading to renewed prominence of the notion of Lawvere theory, now in a computational setting. Here, we investigate the history, in particular asking why Lawvere theories were eclipsed by monads in the 1960's, and how the renewed interest in them in a computer science setting has been developing and might continue to develop in future.
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Khovanov homology is a vector space valued invariant of links whose graded Euler characteristic is the Jones polynomial. It is a stronger invariant than the Jones polynomial, reveals interesting further structure and has nice functorial properties with respect to link cobordisms. In this talk I will endeavour to give an overview of the subject discussing definitions, elementary properties and some applications.
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Wiles' proof of Fermat's Last Theorem is one of the happiest memories of the 20th century. Unfortunately, Wiles' proof does not readily extend in a way that allows us to solve many other classical Diophantine problems. In this talk, based on joint work with Bugeaud and Mignotte, we explain how the proof of Fermat's Last Theorem can be combined with older analytic techniques due to Baker, in a way that solves several classical Diophantine problems. For example, we show that the only perfect powers in the Fibonacci sequences are 0, 1, 8, 144.
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Noncommutative projective geometry seeks to use the intuition and techniques from classical projective algebraic geometry to understand the structure of noncommutative algebras and related modules categories. In this talk I will survey some of the basic ideas and techniques in the subject and, time permitting, outline recent work that describes a large class of ``noncommutative surfaces'' which have some weird and wonderful properties.
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I will introduce the audience to some of the main ideas and methods in non-commutative algebraic geometry by focusing on a rich class of examples, the spaces of the title, and showing how closely their behavior follows that in the commutative case. Our results about non-commutative Hirzebruch surfaces, $qF_n$, specialize to the commutative case: for example, there is a map, in the sense of non-commutative geometry, to the projective line, there is a curve on $qF_n$ with self-intersection number (defined in terms of the Euler form on the Grothendieck group) $-n$, and contracting that curves provides maps to other well-known non-commutative surfaces that are again analogues of their commutative counterparts. The starting point for the definition and analysis is a non-commutative analogue of Cox's homogeneous coordinate ring of a toric variety.
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Hecke algebras arise in various contexts in Mathematics, ranging from knot theory (construction of the famous Jones polynomial) to the theory of finite groups. They possess a rich and involved combinatorial structure. The purpose of the talk is to explain the role that these algebras play in the representation theory of finite groups and to highlight some recent advances.
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The basic questions of diophantine geometry concern rational or integral points on an algebraic variety: do they exist, how can they be described (or found), how are they distributed, etc. Such questions lead to deep theorems (often ineffective in various ways) and far-reaching conjectures. This talk will be about a circle of problems and results on giving simply upper bound estimates for the number of integer or rational points up to a given height. I will describe a quite elementary method that yields results that, while relatively weak for an individual variety, are uniform over large classes of varieties. This uniformity has made the results useful. The same methods are also applicable to certain nonalgebraic sets. I will describe a result about rational points on the graph of a transcendental real-analytic function and a connection with transcendence theory. I will finally describe a result on the rational points of analytic (and more general) sets of arbitrary dimension and further connections with transcendence theory.
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Research into the Baum-Connes conjecture and related issues provides examples of non-trivial and fruitful interactions between analysis and geometry of (among others) discrete groups. On the other hand, ideas of Gromov, Roe and others gave rise to a programme of large- scale geometry, where two objects are declared equivalent if they "look the same" from a distance. In this talk I shall describe new ideas and results that arise from efforts to unify certain features of both programmes.
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Molecular phylogenetics is concerned with inferring evolutionary relationships (phylogenetic trees) from biological sequences (such as aligned DNA sequences for a gene shared by a collection of species). The probabilistic models of sequence evolution that underly statistical approaches in this field exhibit a rich algebraic structure. After an introduction to the inference problem and phylogenetic models, this talk will survey some of the highlights of current algebraic understanding. Results on the important statistical issue of identifiability of phylogenetic models will be emphasized, as the algebraic viewpoint has been crucial to obtaining such results.
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A regular fibration by geodesics of a three-dimensional space form is represented by a spacelike surface in four-dimensional moduli space of geodesics. In the euclidean case, the standard contact structure is perpendicular to such a fibration. Solemn undertaking: "I would make it accessible to any math grad student..."
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This talk will report on a project to understand, extend and consolidate a dense network of connections between a wide range of ideas in stable homotopy theory.
One way into the maze is to consider the symmetric powers of the sphere spectrum, which interpolate between the sphere spectrum itself and the integer Eilenberg-Mac Lane spectrum. The quotients in this filtration are interesting spectra that arise naturally in a number of other contexts, involving the theory of Steinberg modules and Hecke algebras and the combinatorics of partition complexes. The same partition complexes are also relevant in the theory of the Goodwillie tower of the identity functor. There are other connections with power operations in Morava E-theory, as well as the classical Dyer-Lashof algebra and Lambda algebra.
A great deal is already known about these ideas, but there are some hints that important parts of the puzzle have yet to fall into place.
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I will give a geometric characterisation of critical points of eigenvalue functions of the transition probability matrix of random walks on finite, vertex transitive graphs in the case of higher multiplicity. I will also  explain applications to Archimedean solids and to the Cayley graphs of finite Coxeter groups. These are joint results with Ioannis Ivrissimtzis.Â
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We introduce certain algebras of deformed differential operators on a vector space. Their representation theory can be studied via monodromy representations, leading to Hecke algebras. On the other hand, these algebras can be microlocalized. This microlocalization provides a quantization of the Hilbert schemes of points on the complex plane.
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I describe an example, based on a joint work with Paul Monsky, showing that tight closure does not commute with localization. The example is given by a normal hypersurface domain in dimension three in characteristic two and the ideal is generated by three elements. It is a geometric deformation of a two-dimensional tight closure problem and it is analogous to an example of an arithmetic deformation constructed together with Moty Katzman. The geometry in the background of this example is the existence of a vector bundle on a family of smooth projective curves parametrized by the affine line, such that the bundle on the generic curve is strongly semistable, but not so on any special curve. I will also discuss some new developments, discussed with Helena Fischbacher-Weitz, indicating that a certain generic ideal inclusion, which is based on the Froeberg conjecture and which holds for the polynomial ring in three variables, has a tight closure version for graded three-dimensional Cohen-Macaulay domains.
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The class of generalized Weyl algebras was introduced by V.Bavula in early 90s. Quite a lot is known about finitely generated projective modules over GWAs (say their Grotendieck groups and the zeroth cohomology groups were calculated by T.Hodges). In this talk we discuss a recent classification of infinitely generated projective module over GWAs.
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Kostant's problem can be formulated as follows: for which modules M over the universal enveloping algebra U(g) of a semi-simple complex finite-dimensional algebra g does the algebra U(g) surject onto the vector space of all linear endomorphisms of M, which are locally finite with respect to the adjoint action of g. This question is not yet answered even for simple highest weight modules. In the talk we plan to survey the classical results on this problem and present some recent results and applications.
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Deformation theory is the local study of moduli stacks in algebraic geometry. Derived moduli stacks were introduced by Deligne, Drinfel'd and Kontsevich as a generalisation. I will show how strong homotopy algebras and coalgebras over various monads and comonads can be used to construct derived deformation groupoids in many cases.
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Symplectic geometry provides the geometric setting for the Hamiltonian formulation of classical mechanics, and its quantization. Multisymplectic geometry is a generalisation of symplectic geometry which allows an analogous formulation for classical field theory. I will describe this formulation, together with some rather tentative steps towards quantization, with particular reference to functional integrals.
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We prove that the function field of a variety which possesses a special correspondence in the sense of M. Rost preserves the rationality of cycles of small codimensions. This fact was proven by Vishik in the case of quadrics and played the crucial role in his construction of fields with $u$-invariant $2^r+1$. The main technical tools are algebraic cobordism of Levine-Morel, generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. As a direct application of our methods we prove the Vishik's Theorem for all $F_4$-varieties.
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There will be two themes to this talk. One is the general interaction of finite, continuous and algebraic group representation theory, together with related roles of representations of Lie algebras and quantum groups. My point of view will mostly be that of finite and algebraic groups. The second theme is the intereaction of linear and nonlinear actions of these groups and the intereactions of the study of these actions with group cohomology and homological algebra. Finally I will discuss some recent examples and results related to some issues raised by Bob Guralnick on the asymptotic behavior of 1-cohomology for finite groups.
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In this talk, we explore the relationship between integral quadratic forms, both positive definite and indefinite, to modular forms. Modular forms play an increasingly central role in modern number theory. We give an introduction to the subject concentrating on the case of quadratic forms with $3$ variables. Among the topics we discuss are class numbers of imaginary quadratic fields, representation numbers for the sum of three squares, and the values of the famous j-invariant at quadratic irrationalities in the upper half plane.
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The relation between symplectic 2-forms and Poisson structures is well known. We shall show how this relation can be extended to the case of inhomogeneous multivector fields and inhomogeneous differential forms (or pseudodifferential forms). As a starting point we take a transformation from the de Rham complex to the Poisson-Lichnerowicz complex existing on an arbitrary Poisson manifold and, using some analogy with classical mechanics, show how it generalizes to the "non-quadratic" case. The role of inverting a matrix of a 2-form or a contravariant 2-tensor is taken by the Legendre transform. In particular we arrive at a notion which is a generalization of a symplectic structure and gives rise to higher Poisson brackets. We shall discuss homotopy Poisson structures (such a structure makes the space of functions on a manifold an L-infinity algebra) and show how one obtains Koszul type brackets in this setting. The Koszul bracket on an ordinary Poisson manifold is an odd Poisson bracket on the algebra of forms. In particular, it makes the cotangent bundle T*M a Lie algebroid. For a homotopy Poisson structure we give a construction of higher Koszul brackets. The induced structure on T*M will be that of an "L-infinity algebroid". (I shall explain what it means.) The talk is based on a work in progress with H. M. Khudaverdian. See http://arxiv.org/abs/0808.3406.
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The von Neumann algebras (or $W^*$-algebras) were introduced by John von Neumann in 1929 for the needs of quantum mechanics. They make a very important class of operator algebras. There are several equivalent ways of defining them. In particular, von Neumann algebras can be characterized by the existence of the predual, i.e., a Banach space such that the dual of it is the algebra. We shall give an introduction to von Neumann algebras and show that with any von Neumann algebra $M$ one can canonically associate the following structures:
- the structure of a Banach Lie-Poisson space on the Banach space $M_*$ predual to $M$;
- various groupoid structures including a groupoid structure on the set $U(M)\subseteq M$ of partial isometries;
- an inverse semigroup structure on some canonically chosen subsets of $M$.
- A. Odzijewicz, T. Ratiu, Banach Lie-Poisson spaces and reduction, Commun. Math. Phys. 243, 1-54 (2003).
- A. Odzijewicz, A. Sliżewska, Groupoids and inverse semigroups related to a W*-algebra (in preparation).
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We discuss Weyl's character formula, then Lusztig's conjecture. We discuss why it is still open and what tools modern Algebraic Geometry have to facilitate settling it.
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In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical grassmannian (Brown, Goodearl and Yakimov); and torus invariant prime ideals in the quantum grassmannian (Lenagan, Rigal and I). The aim of this talk is to present these results and explore the reasons for this coincidence.
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In 1980, 1981 I had shown that the number of maximal hyperbolic arithmetic reflection groups is finite in each dimension n>9, and in 1981 Vinberg had shown that these groups don't exist for n > 29.
Only in 2005 Long-Maclachlan-Reid proved finiteness in dimension n=2, and Agol in dimension n=3. The remaining gap, 3
Thus, finally, now we know that the number of these groups is finite in all dimensions together.
I plan to outline these results and my further results about effective finiteness (2007). They permit to obtain a finite classification of these groups, in principle.
The groups are important in Algebraic Geometry and in the theory of Lorentzian Kac-Moody (Borcherds) Lie algebras.
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Lie algebroids generalize several geometric objects such as tangent bundles, Lie algebras or Atiyah sequences. It is possible to give a geometric formalism of Lagrangian mechanics on Lie algebroids that particularizes to the usual Euler-Lagrange, Euler-Poincare, or Wong equations for the above particular cases. In this talk we will introduce this formalism and will study the notions of gauge and dynamical equivalence for Lagrangian systems as well as their relationship with Nöther conserved quantities for the dynamics.
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Let $\mathcal N$ denote the set of $n \times n$ nilpotent matrices. The ``enhanced nilpotent cone'' is the space $\mathbb C^n \times \mathcal N$. $GL(n)$ acts on $\mathbb C^n$ in an obvious way, and on $\mathcal N$ by conjugation. The orbits of this action are the subject of this talk. From this surprisingly elementary starting point, I will discuss connections to various topics in representation theory, combinatorics, and algebraic geometry, and especially to Syu Kato's work on the ``exotic nilpotent cone'' and affine Hecke algebras. This is joint work with A. Henderson.
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The multigraded Hilbert scheme, introduced by Haiman and Sturmfels, parameterizes all ideals in a polynomial ring with a fixed multigraded Hilbert function with respect to an abelian group grading. I will discuss joint work with Greg Smith proving the conjecture that when the polynomial ring has two variables these Hilbert schemes are always smooth and irreducible. This reduces to combinatorial commutative algebra.
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I will describe my approach to quantization of Poisson manifolds using symplectic groupoids. By "quantization", I mean the construction of a noncommutative C*-algebra from a geometrical approximation. This approach unifies several previous examples. It combines the ideas of geometric quantization and groupoid convolution algebras. I will try to explain as many of these concepts as time allows.
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Let $G$ be a simple algebraic group defined and split over $\mathbf{F}_p$. We may associate to $G$ the finite groups $G(\mathbf{F}_q)$ for any $p$-th power $q$ and also the Lie algebras $g_{\mathbf{F}_q}$ where $g = Lie(G)$. If $M$ is a rational $G$-module, then $M$ can be viewed as a module for each of the $G(\mathbf{F}_q)$ as well as a "$p$-restricted" representation for each of the $g_{\mathbf{F}_q}$. Work of J. Carlson, Z. Lin and D. Nakano relates cohomological invariants associated to $M$ as a $G(\mathbf{F}_p)$-module and as a $g$-representation. In this talk, we describe the Weil restriction functor which converts structures over $\mathbf{F}_q$ to structures over $\mathbf{F}_p$. We apply this functor to the Seitz log map relating unipotent and nilpotent elements in order to formulate and prove various cohomological results for $G(\mathbf{F}_q)$.
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An important problem in algebra is the study of algebraic objects defined over fields and how they behave under field extensions, for example the Brauer group of a field, Galois cohomology groups over fields, Milnor K-theory of a field, or the Witt ring of bilinear forms over a field. Of particular interest is the determination of the kernel of the restriction map when passing to a field extension. We will give an overview over some known results concerning the kernel of the restriction map from the Witt ring of a field to the Witt ring of an extension field. Over fields of characteristic not two, general results are rather sparse. In characteristic two, we have a much more complete picture. In this talk, I will explain the full solution to this problem for extensions that are given by function fields of hypersurfaces over fields of characteristic two. An important tool is the study of the behaviour of differential forms over fields of positive characteristic under field extensions. The result for Witt rings in characteristic two then follows by applying earlier results by Kato, Aravire-Baeza, and Laghribi. This is joint work with Andrew Dolphin.
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Understanding conductors of elliptic curves via geometric adelic two-dimensional theory Short abstract: The classical way of working with arithmetic aspects of elliptic curves over a global field is to involve (generally noncommutative) Galois extensions of the (one-dimensional) global field. In this classical approach the conductor of elliptic curve remains a mysterious object. Geometrically, elliptic curves over global fields can be viewed via associated two-dimensional models. I will try to explain how the recent theory of two-dimensional adelic spaces and zeta integrals on them gives a real understanding of the conductor (at least, its tame part) of elliptic curves, which is not available in the classical one-dimensional theory.
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Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders 3-dimensional Calabi Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded category algebras with cancellation. For this broader class the CY-3 condition also implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold. We discuss which of these quiver polyhedra give rise to Calabi Yau algebras.
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This is not really what I did on my study leave, but during my study leave I had this 3 page paper published. While the result is truly insignificant, it gives me a good excuse to talk about elliptic curves and modular curves in colloquium style, and saying something about the proof will provide a focus for the discussion.
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Everyone has probably seen a proof that the higher homotopy groups of a space are abelian that consists of a series of pictures in which squares slide around one another. This is one example of the Eckmann-Hilton argument, which is a very simple piece of algebra that allows one to conclude that a group (or more generally, monoid) is abelian. I will explain the algebra involved, give some nice examples from topology, and then show that the situation becomes much more complicated when dealing with examples from category theory.
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My talk will be based on joint work with Claude LeBrun, mostly in arxiv:0806.3761. The dicussion is motivated from a class of integral formulae for solutions to the wave equation in 2+1 dimensions. The question of what spaces admit such integral formulae is addressed and it turns out that the wave equations are most naturally defined on Einstein-Weyl spaces. When subject to a suitable global assumption, these Einstein-Weyl spaces are classified by a scattering map, a smooth diffeomorphism from the two-sphere at past infinity to one at future infinity along null geodesics. The Einstein-Weyl space is then reconstructed from a family holomorphic discs in an auxilliary complex surface with boundary defined by the scattering map. If I have time I will discuss more recent applications to the classification of Zoll surfaces, manifolds all of whose geodesics are closed circles.
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In his manuscripts from the 1980's Grothendieck proposed ideas that have been interpreted variously as embedding the theory of schemes into either -group theory and higher-dimensional generalizations; -or homotopy theory. It was suggested, moreover, that such a framework would have profound implications for the study of Diophantine problems. In this talk, we will discuss mostly the little bit of progress made on this last point using some mildly non-abelian motives associated to hyperbolic curves.
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We describe the solution to this long-open problem in representation theory.
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Classical knot theory is the study of embedded circles in 3-dimensional space. The purpose of this talk is to illustrate the rich give-and-take between knot theory and 4-dimensional topology. I will discuss the use of knots in descriptions of 3- and 4-dimensional manifolds. I will also describe how a 4-dimensional point of view of knots gives rise to a group called the knot concordance group, and discuss some recent advances in the study of this group.
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We discuss a proof of Benson's regularity conjecture, that the regularity of the cohomology of a finite group is zero. The regularity is an invariant defined in term of local cohomology and knowing it gives bounds on the degrees of the generators and relations of the ring. The proof is a mixture of algebra and topology.
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My talk will be a survey of assorted work on spaces of group actions on trees and some associated completions and deformations of group algebras. Such a space was defined by Culler and Vogtmann in the study of outer automorphisms of free groups and later linked in with Kontsevich's graphical calculus. At the opposite extreme such spaces have been useful in the representation theory of crossed products by free abelian groups. I am also expecting them to arise in the theory of Iwasawa algebras, (completed) group algebras of p-adic Lie groups.
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The classical Hasse principle states that an abelian group M can be recovered from its rationalization and its p-adic completions for all primes p, in the sense that if M is finitely generated, there is a suitable pullback square recovering M. This arithmetic principle applies to modules over many other commutative rings, to sheaves over an elliptic curve and to circle-equivariant cohomology theories. I intend to explain the idea behind this and some classification theorems that follow from it.
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The notion of a Maurer-Cartan (MC) element in a differential graded Lie algebra is an abstraction of the notion of a flat connection on a vector bundle. MC elements and their moduli spaces appear in disparate areas of mathematics; usually in the context of deformations of various types of objects (complex-analytic structures, connections, associative and Lie algebras and their homotopy invariant versions etc.) I will give a modern overview of MC theory from the point of view of L-infinity algebras and describe how one can twist structures by an MC element. As an example of the general technology I will describe a canonical L-infinity map from an A-infinity algebra to its Hochschild complex and, if time permits, outline an application to graph cohomology. This is joint work with J. Chuang.
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One of the most fundamental theorems in representation theory is the Beilinson-Bernstein theorem, which describes the representations of a Lie algebra such as sl_n in much simpler terms---as sheaves of modules for differential operators on a smooth algebraic variety. I will describe this theorem, and some work in progress to generalise it to describe non-commutative deformations of enveloping algebras in terms of 'almost local differential operators'.
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Relative algebraic geometry can be done over a symmetric monoidal category. This gives a way of doing algebraic geometry over the field with one element. One can use Kashiwara's crystal bases to define reductive groups and Schubert varieties over the field with one element. These are related to toric degenerations of Schubert varieties. This is joint work with Nick Royzenblum.
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There are a number of situations in mathematics where a C*-algebra is associated to a geometric object. Often, this association is not functorial. In this talk we look at a generalisation of C*-algebras called C*-categories, and show how, in many situations, a C*-algebra can be replaced by an equivalent C*-category- and in the C*-category setting, the association we obtain is functorial.
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Finite-dimensional algebras with trace forms give rise to topological invariants of Riemann surfaces. I will describe joint work with Andrey Lazarev on a variation due to Kontsevich that produces better invariants.
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Juergen Herzog, in a by now classic paper, gave minimal generating set for the vanishing ideal of the affine space curve $% k[t^{n_{1}},t^{n_{2}},t^{n_{3}}]$ under the restriction that the $n_{i}$ form a relatively prime triple of positive integers. Necessarily this ideal is prime, and is either a complete intersection or an almost complete intersection with generators the $2\times 2$-minors of the matrix \begin{equation*} \left( \begin{array}{ccc} x^{a_{1}} & y^{a_{2}} & z^{a_{3}} \\ y^{b_{2}} & z^{b_{3}} & x^{b_{1}} \end{array} \right) \end{equation*} where the $a_{i}$ and $b_{j}$ are positive integers. We used these ideals quite recently to answer in the negative a twenty year old problem about the Uniform Artin-Rees property on the prime spectrum of an excellent ring. In this talk we discuss the properties of this ideal in general. We first show that it is a Northcott ideal (in the sense of Vasconcelos). This enables us to display immediately its properties connected to liaison and the fact that it is unmixed. Next we revisit work of Herzog, Bresinsky and Valla connected with the property of being a set-theoretic complete intersection. Finally we focus on the case of the polynomial ring $k[x,y,z].$ We show that the ideal is prime only in the case treated by Herzog, that it is usually radical, and we use the theory of multiplicities to estimate the number of its components. Some extensions of aspects of this work will be sketched.
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To any nodal rational projective curve we associate a non-commutative curve called its ``Auslander curve.'' We study its homological properties and construct a tilting sheaf, which establishes an equivalence of the derived category of coherent sheaves over such a curve and that of modules over a finite dimensional algebra. We also study the embedding of the category of coherent sheaves over the initial nodal curve into the category of coherent sheaves over its Auslander curve.
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The local Langlands conjectures predict a natural surjection from the set of equivalence classes of irreducible smooth complex representations of a p-adic classical group G to the continuous representations of the Weil-Deligne group in the Langlands dual group of G. The fibres of this map are the L-packets of the title. This map is known to exist for Sp(4), from work of Gan--Takeda, and now in general, from work of Arthur to appear. However, this still leaves the question of understanding the L-packets explicitly. As well as explaining this background in more detail, I will report on work in progress with Corinne Blondel and Guy Henniart giving an approach to finding these L-packets explicitly, using Bushnell--Kutzko's theory of types and work of Moeglin.
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Given a one-dimensional Cohen-Macaulay local ring we study and compare several families of invariants related with its tangent cone. When $A$ is equicharacteristic, Juan Elias introduced in $2001$ the set of micro-invariants of $A$ in terms of its first neighborhood ring. On the other hand, if $A$ is a one-dimensional complete equicharacteristic and residually rational domain, Valentina Barucci and Ralf Fröberg defined in $2006$ a set of invariants in terms of the Apery set of the value semigroup of $A$, and showed that both sets of invariants coincide if the tangent cone $G(A)$ is Cohen-Macaulay. We give a new interpretation for these sets of invariants that allow to extend them to any one-dimensional Cohen-Macaulay local ring. We compare these sets with the family of invariants recently introduced by Teresa Cortadellas and the speaker for the tangent cone of a one-dimensional Cohen-Macaulay local ring and give explicit formulas relating them. We show that, in fact, they coincide if and only if the tangent cone $G(A)$ is Cohen-Macaulay. Some explicit computations will also be given, particularly for the case of semigroup rings.
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The three-sphere recognition problem dates to Poincare's pioneering work in ``analysis situs''. The unknot recognition problem is even older and in some sense predates topology itself. I will give an elementary introduction to the work of Haken and Rubinstein on these problems, ending with a discussion of their computation complexity.
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A survey of the relationship between the geometry of varieties associated to algebraic groups and representation theory of the group - mostly historical but, in particular mentioning the solution of Hilbert's 14th problem (in invariant theory of reductive groups) but coming up to date with some concrete calculations of cohomology groups of line bundles via representation theory.
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A lot of progress in algebraic geometry has been made from the discovery that the derived category of sheaves on a variety describes the branes in a particular topological field theory associated to that variety. I'll try and explain roughly what all of these words mean, and then describe how a more general field theory leads to a generalization of the derived category. I'll then tell you some of the things you can do with this generalization, and in particular how it links to the matrix factorizations studied in algebraic singularity theory.
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We all learned about induction for proving properties of natural numbers at school. But what about other data types such as lists -- do they have induction principles? And what exactly are properties and can we use induction for different notions of properties. Finally, if we work in categories other than the category of sets, do we still have induction principles? In this talk I'll show that, indeed, we can have induction principles parameterised by the category we work in, the data type we are interested in and the notion of property under consideration. These results arise from a very beautiful picture of induction if one will willing to consider the topic within a fibrational setting.
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Since Euler we know that the value of the Riemann zeta function at non-positive integers is a rational number. Do these values have any meaning? The Bloch-Kato conjectures answer this question more generally for values of so-called L-functions. I will give an introduction to the conjectures and explain a strategy first employed by Ribet that continues to yield new results towards them.
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I will try to give an overview of the big machine called "categorification" with emphasis on applications in representation theory. Over the last thirty years it has been observed that interesting algebras (group algebras, Hecke algebras, enveloping algebras) act on Grothendieck groups of categories arising in representation theory. If one asks what structures act on the categories themselves one is led to consider categorifications of these algebras. It turns out that these categorifications often have a very rigid structure and lead to new insights in representation theory.
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We give a non-technical introduction to differentiable/topological stacks and discuss some old and new results. Along the way we provide plenty of examples and give some concrete applications of these results.
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In classical surface theory, transformations have been used to locally construct surfaces out of given simple ones by replacing a PDE which describes a certain surface class by an ODE or a simpler PDE which describes the transformation. In my talk, I will explain two transformations of constant mean curvature surfaces: the simple factor dressing is linked to a harmonicity condition, whereas the Darboux transformation uses that the surface is isothermic. In the case of constant mean curvature surface these two transformations are the same; in particular, these transformations also give global information, e.g., points of the spectral curve of a constant mean curvature torus are given by simple factor dressing of the torus.
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Elliptic curves $E$ and $F$ are $n$-congruent if their $n$-torsion subgroups are isomorphic as Galois modules via an isomorphism respecting the Weil pairing. Rubin and Silverberg gave formulae for the families of elliptic curves $n$-congruent to a given elliptic curve for $n=2,3,4,5$. Formulae in the case $n=7$ are given by Halberstadt and Kraus. I will describe an invariant-theoretic approach to obtaining these formulae and use it to extend to the cases $n=9$ and $n=11$. There are corresponding formulae when the isomorphism of $n$-torsion subgroups does not respect the Weil pairing.
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This talk is about random matrices. If we select an $n \times n$ orthogonal matrix $X$ ``at random'', using the uniform distribution on the orthogonal group $\mathrm{O}(n)$, then the powers of $X$ are not uniformly distributed in $\mathrm{O}(n)$. However, as $n$ increases, the distribution of $X^n$ stabilizes. We prove this, consider generalizations to matrices in other compact Lie groups, and make some remarks about random matrices in other Lie groups.
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I will report on some recent joint work with Robert Gray, which aims to extend geometric methods from group theory to semigroups and monoids. The talk will start with an introduction to geometric group theory, and should be accessible to a general mathematical audience.
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The underlying theme of the talk is the relationship, via quantization and semiclassical limits, between Poisson algebras and parametrized families of noncommutative algebras. I intend to discuss this through a case study beginning with a nonlinear recurrence sequence arising in work of Fordy and Marsh on periodic quiver mutation. Understanding the recurrence leads to a Poisson bracket on a rational function field, with a Poisson automorphism modelling the recurrence, and to an affine Poisson subalgebra that establishes complete integrability for the automorphism. This leads in turn to my current desert island ring, a noncommutative deformation of this Poisson subalgebra. Although the recurrence is nonlinear it can be linearized in a simple way and this idea carries forward throughout the rest of the talk.
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The title is dull, but the material is not. Given a mathematical structure it is usually straightforward to define a good notion of morphism. In the case of Lie algebroids, this only happened about 25 years after their introduction, and when a definition was given, it was widely disliked. The main purpose of the talk is to display the range of differential geometric phenomena which are covered by the concept: Maurer-Cartan forms, symplectic realizations, Poisson sigma models, as well as the more obvious ones.
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The classical theorems of Geometric Topology relate geometric, topological and analytic invariants of a compact manifold. These include the Gauss-Bonnet Theorem, the Hodge Theorem, the Hirzebruch Signature Theorem and the Atiyah-Singer Index Theorem. These theorems are interesting and important, because they allow information to travel among the areas of geometry, topology and global analysis. After the proof of the Index Theorem in the 70's, there was interest in extending these remarkable results to classes of manifolds with some sorts of singularities. However, the natural direct generalisations are not correct. In this talk, I will start by reviewing the central results in the compact setting. Then I will outline some of the developments in topology and in global analysis that have given us the language in which to express relationships between analysis and topology in the singular setting. Finally, I will give an overview of the progress so far on this general project.
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The subject area of my talk can be classified as psychology of mathematical abilities, but approached from a mathematician's point of view. I will discuss some hidden structures of elementary school mathematics (frequently quite sophisticated and non-elementary) and conjectural cognitive mechanisms which allow some of so-called "mathematically able" children to feel the presence of these structures.
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In 1968, Dixmier posed six problems/conjectures for the algebra of polynomial differential operators, i.e. the Weyl algebra. In 1975, Joseph solved the third and sixth problems and, in 2005, I solved the fifth problem and gave a positive solution to the fourth problem but only for the homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra 'behaves' like a finite field extension. In more detail, the first problem/conjecture of Dixmier asks: is it true that an algebra endomorphism of the Weyl algebra an automorphism? In 2010, I proved that this question has an affirmative answer for the algebra of polynomial integro-differential operators. In my talk I explain the main ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.
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Quiver mutation is an elementary operation on quivers which appeared in physics in Seiberg duality in the 1990s and in mathematics in Fomin-Zelevinsky's definition of cluster algebras in 2002. In this talk, I will show how, by comparing sequences of quiver mutations, one can construct identities between products of quantum dilogarithm series. These identities generalize Faddeev-Kashaev-Volkov's classical identity and the identities obtained recently by Reineke. Morally, the new identities follow from Kontsevich-Soibelman's theory of refined Donaldson-Thomas invariants. They can be proved rigorously using the theory linking cluster algebras to quiver representations.
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I will describe how two seemingly different constructions are examples of a more general construction. The first construction, the tight span of a metric space, is a way of possibly embedding a finite metric space in a tree, and has applications in biological evolutionary trees and in network theory. The second construction, the Dedekind-MacNeille completion of a poset, is familiar as one way of completing the rational numbers to the real numbers, via Dedekind cuts. I will then explain how these are two examples of a construction from cateogory theory and say how this allows useful generalization.
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Random matrix theory has played a huge role in the last decade in aiding the understanding of the distribution of $L$-functions. This talk will survey some of that material, and look forward to newer results on their extreme values.
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During a series of lectures titled ``Old and new problems in physics'' delivered to the faculty of the University of Göttingen in 1910 (with David Hilbert and his student Hermann Weyl in the audience), the Dutch physicist, Hendrik Antoon Lorentz, conjectured that the number of eigenvalues for the Laplacian (under suitable boundary conditions) for a region of area $A$ that do not exceed the positive number $n$, is proportional to $A$ times $n$, when $n$ is large. While (apparently) Hilbert predicted that the conjecture would not be proved in his lifetime, the result was proved by Weyl in 1912. From this point on, Weyl's celebrated theorem (commonly referred to as Weyl's Law) has been studied, extended, generalized, etc., by mathematicians and physicists in many different settings. After recalling some of the classical results we will look at similar problems when the Brownian motion, which "goes'' with the classical Laplacian, is replaced by other Lévy processes, in particular the rotationally invariant stable processes which "go" with fractional powers of the Laplacian.
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Modules are usually thought of as sets with structure but they may also be seen as functors, indeed by increasing the exactness requirements on these functors we are led to quite different views of modules and, from there, to close links between various kinds of additive categories. These links can be expressed as (anti-)equivalences between certain 2-categories. I will describe this and also mention how this picture in the additive context parallels one, which involves toposes, in the Set-based context.
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In a polynomial ring, a binomial says that two monomials are scalar multiples of each other. Forgetting about the scalars, a binomial ideal describes an equivalence relation on the monoid of exponents. Ideally one would want to carry out algebraic computations, such as primary decomposition of binomial ideals, entirely in this combinatorial language. We will present a calculus that enables one to carry out algebraic computations by "looking at pictures".
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Cohomology theories are of great importance for studying topological spaces. They take as input topological spaces and as output give a collection of abelian groups. They satisfy a useful collection of axioms which (in theory) make these groups computable. If $X$ is a space with an action of a compact Lie group $G$ and $E$ is a cohomology theory, then $E^*(X)$ also has a $G$-action. But this doesn't usually tell us a great deal about the action, for example if $G$ is the circle group, then the action on cohomology is always trivial. So there is a need for cohomology theories that use the $G$-actionin a more fundamental way. These are called equivariant cohomologytheories, we discuss the definition, consider some examples and show that rationally, they are very well-behaved.
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I will describe Ramanujan's mod 691 congruence and its proof, then describe an instance of a conjecture about related congruences, and the search for numerical evidence.
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Quantized enveloping algebras are Hopf algebras that are q-deformations of universal enveloping algebras. Despite being defined by a bunch of peculiar looking relations, they have found applications in many parts of maths and physics. Twenty years ago Ringel showed how to give a conceptual description of the positive half of a quantized enveloping algebra using Hall algebras of quiver representations. I'll attempt to explain why introducing Z_2 graded complexes into the picture leads to a similar description of the whole thing.
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The mathematical theory of the shape of objects is still in its infancy - as yet there is no "language" of shape. In this talk I will consider different approaches via differential geometry to the problem of shape description and show how one in particular, involving the Laplace-Beltrami operator, captures the geometry of a shape. Since in practice shapes are defined by discrete data such as a triangulation or point cloud, I shall show how one can, via a discrete version of exterior calculus, construct a discrete version of the Laplace-Beltrami operator, the eigenvalues and eigenfunctions of which encode an immense amount of useful information about the shape. Applications can be made to a variety of situations, eg leaf shape, medical imaging.
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In this talk we study a quadratic Poisson algebra structure on the space of bilinear forms on $\mathbb C^{N}$ with the property that for any $n,m\in\mathbb N$ such that $n m =N$, the restriction of the Poisson algebra to the space of bilinear forms with block upper triangular matrix of $n^2$ blocks of size $m\times m$ is Poisson. We characterise all central elements of this Poisson algebra and construct the braid-group action that preserves the Poisson algebra on each Poisson restriction. If time allows, we will discuss quantisation.
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Quantized enveloping algebras are Hopf algebras that are $q$-deformations of universal enveloping algebras. Despite being defined by a bunch of peculiar looking relations, they have found applications in many parts of maths and physics. Twenty years ago Ringel showed how to give a conceptual description of the positive half of a quantized enveloping algebra using Hall algebras of quiver representations. I'll attempt to explain why introducing $Z_2$ graded complexes into the picture leads to a similar description of the whole thing.
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Dehn surgery is a classical area of low dimensional topology with many beautiful results connecting the subject matter to the description of 3-manifolds, the original Poincare conjecture, and the geometry of knot exteriors. In this talk we will introduce and motivate Dehn surgery with a view to speaking about ``exceptional surgeries"; this will naturally bring us to a well known tabulation of 3-manifolds of ``small volume". It will be our goal to discuss an unusually simple description of the ``exceptional fillings" associated with this tabulation. The presentation will attempt to be intuitive and contain many pictures.
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In this talk, I will begin to give a brief review on the algebraic or p-adic aspect of modular forms. Then I will move on to the modern view of modular forms with its relation to Iwasawa theory. If time permits, I would like to mention some recent topics. This talk is elementary.
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I will describe a relationship between certain monoidal categories called fusion categories and 3-dimensional topological field theories, focusing on the correspondence between algebraic properties of the categories and topological properties of the associated field theories. Fusion categories are monoidal categories that have the nice properties of the category of representations of a finite group: each object has a dual, there are finitely many simple objects, and any object decomposes into a finite sum of simples. We show that any fusion category gives rise to a 3-dimensional topological field theory. A key question about the algebraic structure of a fusion category is whether the double dual operation is trivial, as it is in the representation category of a finite group. I will explain why this question corresponds to the question of whether the 3-manifold invariants of the associated field theory depend on a spin structure. This is joint work with Chris Schommer-Pries and Noah Snyder.
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In the 1950s Jacques Tits formulated a conjecture about spherical buildings, which he had recently invented. In the intervening years, several important special cases of this conjecture have been proved, but the full conjecture is still open. In this talk I will explain what spherical buildings are, from a variety of different viewpoints, and what Tits's conjecture says about their structure. I'll illustrate the various known cases with some straightforward examples which shouldn't need more than a smattering of linear algebra and some geometric intuition. If time permits I'll also detail my small contribution to the effort to prove the full conjecture.
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The theory of complex multiplication allows one to construct elliptic curves with a given number of points. The idea is to construct a curve over a finite field by starting with a special curve $E$ in characteristic $0$, and taking the reduction of $E$ modulo a prime number. Instead of writing down equations for the curve $E$, one only needs the minimal polynomial of its $j$-invariant, called a Hilbert class polynomial. The coefficients of these polynomials tend to be very large, so in practice, one replaces the j-invariant by alternative 'class invariants'. Such smaller class invariants can be found and studied using an explicit version of Shimura's reciprocity law. The theory of complex multiplication has been generalized to curves of higher genus, but up to now, no class invariants were known in this higher-dimensional setting. I will show how to find smaller class invariants using a higher-dimensional version of Shimura's reciprocity law.
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Examples of platonic polygonal complexes include the five regular solids, and the tesselations of the Euclidean and hyperbolic planes by regular polygons. There appear to be too many of them to hope to classify them all, but there are good results for some subfamilies. I shall state some of these results, and explain why the case when the polygons have at least six sides is the simplest case. [This is a self-contained talk, but if you enjoy this then you might be interested in the topology seminar on Thursday which will follow on from it.]
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Do infinitesimals exist? Bearing in mind that Pete and Bernie's Philosophical Steakhouse is now closed\footnote{http://www.youtube.com/watch?v=6Za2WFVrvpQ#t=275s}, I'll discuss two approaches to a non-standard system of analysis which starts from the premise that maybe they should. The original approach due to Abraham Robinson (1960s) has model theory as its basis and I'll cover this set-up from scratch. A reworking due to Edward Nelson (1970s) is based on an extension of ZF set theory and I'll discuss similar ground from this alternative viewpoint, where a possible conclusion is that infinitesimals were always knocking around, we just didn't notice them.
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I shall give an introduction to the cohomology of (mostly finite) groups for a general mathematical audience, from the points of view of homological algebra, topology, commutative algebra, representation theory, and (time permitting) number theory.
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A lattice is a discrete subgroup of a Lie group with finite covolume (with respect to Haar measure). Often the discreteness follows (in a rather general and abstract sense) from the discreteness of the integers in the reals, in which case the lattice is said to be arithmetic. In this talk I will survey lattices and arithmetic groups; constructions of non-arithmetic lattices and an ongoing project to produce new examples.
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The classical localization theorem of Beilinson and Bernstein shows that the representation theory of semisimple Lie algebras can be studied geometrically via modules for the sheaf of differential operators on the flag variety. Recently there has been much interest understanding other contexts in which a similar localization results hold. We will review the classical theory and explain some of the recent developments.
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Any smooth surface of genus $g>1$ embedded in $S^3$ has a canonical structure as a Riemann surface. It can thus be expressed as a branched cover of the Riemann sphere, or as the quotient of the open unit disc by the action of a Fuchsian group. This is a remarkably rich structure, but the literature does not seem to contain any examples where it can all be made explicit. In this talk we will describe a certain highly symmetric surface with many interesting features, where we are close to finding parametrizations of the types described.
Some computations for this project were carried out by Gemma Halliwell as a summer research project, supported by a Burkill studentship.
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A major part of the Langlands programme is a conjectural relationship between number-theoretic objects, such as Galois representations, and analytic ones, for example automorphic forms. These conjectures are now mostly proved in the context of classical modular forms on the one hand, and two-dimensional representations of Galois groups over Q on the other. One of the key results is Serre's Conjecture, proved by Khare and Wintenberger, which can be viewed as a mod p Langlands correspondence. An important feature of Langlands correspondences is a local-global compatibility principle, but it is not even known how to formulate a conjectural mod p version of this principle beyond the context of classical modular forms. I'll discuss what's known and what some of the difficulties are.
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The Dependent Type Theory of Per Martin-Lof is one of the most remarkable developments in Logic of the last 50 years; but its significance is difficult to pin down. It was taken up very early as the basis both for programming and specification languages and it stimulated work in theoretical computer science. However little serious attention was paid to one striking feature, the identity types. These are a clear pointer to a connection with basic ideas of homotopy theory, but it was not evident how to exploit this idea. Recently Voevodsky has made proposals for so-called Homotopy Type Theory and the outlines of a mathematical theory are beginning to emerge. I shall try to make the underlying ideas accessible to all.
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A-infinity structures arise in topology to describe a multiplication which is "associative up to homotopy". They have become important in various different areas of mathematics, including algebra, geometry and mathematical physics. After a brief survey, my main emphasis will be on the theory of minimal models. This involves studying differential graded algebras (dgas) via A-infinity structures on their homology algebras. I will also discuss a recent generalisation of these ideas.
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After a brief introduction of vertex algebras in general, I will describe an explicit example (the Weyl vertex algebra) related to the loop space of $R^n$. When $R^n$ is replaced by a more general manifold $M$, the construction encounters an obstruction, and provides a partial formulation of an important quantum field theory associated to $M$. This construction was first given by patching local data over coordinate charts, but there is also a `cleaner' method using principal bundles and semi-infinite cohomology. In fact, the latter is a special case of a more general construction that should have an interpretation in terms of the loop space of $M$.
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An Artin group is a group with a presentation of the form $$ \langle x_1,x_2,\cdots,x_n \mid x_ix_jx_i\cdots(m_{ij} \text{ terms })= x_jx_ix_j \cdots (m_{ij} \text{ terms }), i,j \in \{1,2,\cdots,n\}, i\neq j\rangle$$ for $m_{i,j} \in \mathbb{N} \cup \infty, m_{ij} \geq 2$, which can be described naturally by a Coxeter matrix or graph. This family of groups contains a wide range of groups, including braid groups, free groups, free abelian groups and much else, and its members exhibit a wide range of behaviour. Many problems remain open for the family as a whole, including the word problem, but are solved for particular subfamilies. The groups of finite type (mapping onto finite Coxeter groups), right-angled type (with each $m_{ij} \in \{2,\infty\}$), large and extra-large type (with each $m_{ij}\geq 3$ or $4$), FC type (every complete subgraph of the Coxeter graph corresponds to a finite type subgroup) have been particularly studied. After introducing Artin groups and surveying what is known, I will describe recent work with Derek Holt and (sometimes) Laura Ciobanu, dealing with a big collection of Artin groups, containing all the large groups, which we call `sufficiently large'. For those Artin groups we have elementary descriptions of the sets of geodesic and shortlex geodesic words, and can reduce any input word to either form. So we can solve the word problem, and prove the groups shortlex automatic. For many of those groups we can deduce the rapid decay property and verify the Baum-Connes conjecture. I'll explain some background for these problems, and outline their solution.
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In 3- and 4-dimensional hyperbolic spaces there exist regular tilings/mosaics (honeycombs). A belt can be created around an arbitrary base vertex of a mosaic. The construction can be iterated and a crystal-growing ratio can be determined by using the number of the cells of the considered belts. In my talk I would like to introduce some hyperbolic regular mosaics and determine their crystal-growing ratios.
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I will review well-known parts of the classification of complex algebraic varieties in low dimension, starting with curves, which are classified into irreducible families by their genus, and surfaces, where additional invariants come into play. My aim is to survey the extent to which we know all Fano 3-folds, which are the (complex) 3-dimensional analogues of curves of genus zero. I will draw a map (that is, a region in a space of integer invariants like the genus) where they all must live, and explain recent attempts to populate this. In particular, I will sketch a method that constructs different families of varieties that have the same basic algebro-geometric invariants (their associated commutative Gorenstein coordinate rings have the same Hilbert series) but are distinguished by their topology (they have different Euler characteristic).
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Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and, if time permits, for certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.
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I will define "meta-groups" and explain how one specific meta-group, which in itself is a "meta-bicrossed-product", gives rise to an "ultimate Alexander invariant" of tangles, that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.
All the terms used in the above paragraph will be defined during my talk.
Please see http://www.math.toronto.edu/~drorbn/Talks/Sheffield-130206/ for further information and a handout.
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Degenerate $n$-categories are those whose lowest $k$ dimensions are trivial, for some $k>0$. These can be thought of as the categorical analogue of loop spaces. The resulting multiplicative structures are interesting in their own right, and fit into a table known as the ``Periodic table of $n$-categories''. The table has various interesting patterns observable in low-dimensions and conjectured in general by Baez and Dolan. The structures that arise in this way include monoids and commutative monoids, as well as monoidal, braided, and symmetric monoidal categories. We will outline the main ideas behind the Periodic Table, its patterns and predictions, including the crucial stabilisation property which lead Baez and Dolan to conjecture a beautiful universal property to characterise higher-dimensional tangles. The talk will be introductory, and knowledge of definitions of $n$-category will not be assumed.
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One simple chemical model for stability of benzenoids uses the Fries number (the maximum number of benzenoid hexagons over all perfect matchings (a.k.a Kekule structures) of the molecular graph. A recently published algorithm purports to find perfect matchings realising the Fries number of a general benzenoid molecule. We show that this algorithm does not always work correctly, and discuss the chemical significance of what the algorithm is 'really' doing and why it might be useful in spite of its faults. This talk is based on joint work with Wendy Myrvold, Dept. of Computer Science, University of Victoria
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I will discuss some recent developments in Schubert calculus and a potential relation to classical combinatorics for symmetric groups and possible extensions to complex reflection groups.
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The ring of invariants for the action of the automorphism group of the projective line on the n-fold product of the projective line is a classical object of study. The generators of this ring were determined by Kempe in the 19th century. However, the ideal of relations has been only understood recently in work of Howard, Millson, Snowden and Vakil. They prove that for n>6, the ideal of relations is generated by quadratic equations using a degeneration to a toric variety. I will report on joint work with Benjamin Howard where we compute the Hilbert functions of these rings of invariants, and further study the toric varieties arising in this degeneration. As an application we show that the second Veronese subring of the ring of invariants admits a presentation whose ideal admits a quadratic Gröbner basis.
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(Joint work with A.Braun) An ideal $I$ of a ring $R$ is said to have the right Artin-Rees (AR) property if for every right ideal $E$ of $R$ there exists a positive integer $n$ such that $E \cap I^n \subseteq EI$. Unlike in the commutative case, the AR property does not hold universally for all ideals of a non-commutative Noetherian ring. For instance, for prime ideals (some form of) the AR property is intimately related to their localisability. We discuss the longstanding problem of whether the Jacobson radical of a prime Noetherian ring has the AR property.
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In this talk, I will explain how one can use tools develop to study prime ideals of noncommutative algebras in order to study totally nonnegative matrices.
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The Riemann zeta function can be defined first as an infinite sum, and then (by analytic continuation) as a function on the whole complex plane, with a pole at $s=1$. Facts about the zeta function sometimes imply facts about prime numbers and arithmetic, and if we knew the Riemann Hypothesis we would know even more about prime numbers than we do now. The zeta function is an inherently interesting object, and it is not unsurprising that there are now many generalisations of it, encoding other secrets about mathematics. I'll talk about it and its generalisations, so-called $L$-functions, and try to give some idea about what we know and what we want to know about them.
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Suppose M is a 3 or 4 dimensional manifold, and that x is an element of H_2(M). What is the minimal genus of an embedded orientable surface representing x? I'll discuss what we know and don't know about this question, and what it can tell us about topology in dimensions 3 and 4.
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You know about representations of groups, and you've heard about representations of quivers. In this talk I will talk about representations of manifolds. Indeed, compact oriented manifolds of dimensions 1, 2 and 3 organize themselves into a structure known as a "bicategory", and an oriented 123 topological quantum field theory (TQFT) is a representation of this structure. They turn out to be classified by structures known as "anomaly-free modular categories". I will describe my recent work in this area, with examples. Along the way we will stumble across Morse theory, group cohomology and quadratic forms. Joint work with Jamie Vicary, Chris Schommer-Pries and Chris Douglas.
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A good way of understanding groups is to look at its representations into the group of invertible n by n invertible matrices. One can organise such representations into families by first fixing a representation into $GL_n$ of a finite field and then lifting it to `bigger' rings. By a theorem of Mazur, under certain hypothesis the liftings fit into a nice universal family. I will discuss the inverse problem of realizing rings as the coefficient rings of such a universal family and results in this direction (joint work with Tim Eardley).
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Over the last 10 years, the classification of algebraic varieties has seen dramatic progress via the success of the minimal model program (MMP). MMP in turn is based on notions of positivity (of divisors) that fundamentally characterize algebraic geometry. In the talk, I will describe a new approach towards understanding positivity of divisors via the derived category and Bridgeland stability conditions; this is based on joint with with Emanuele Macrì. Via this approach, we can make general MMP-existence results effective, and on the way answer many concrete questions on the geometry of Hilbert schemes of K3 surfaces.
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A differential graded algebra is a chain complex with a multiplication that is compatible with the differentials. This means that its homology also carries an algebra structure. But how many differential graded algebras realise the same homology algebra? We explain why this question is relevant to topology and present an example from homotopy theory.
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Harmonic spaces are Riemannian manifolds on which all harmonic functions satisfy the mean value property. The Lichnerowicz conjecture stated that all simply connected harmonic manifolds are flat or rank-1 symmetric spaces. Szabo proved this conjecture in the compact case in 1990. Shortly afterwards in 1992, there appeared non-compact nonsymmetric harmonic spaces - the so-called Damek-Ricci spaces - disproving the conjecture in the noncompact case. In this talk I will introduce harmonic spaces and the more general asymptotically harmonic spaces and discuss some recent results about these interesting manifolds.
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I will describe an efficient method for computing period lattices and elliptic logarithms for elliptic curves defined over C, using the complex Arithmetic-Geometric Mean (AGM) first studied by Gauss. Previous work has only considered the case of elliptic curves defined over the real numbers; here, the multi-valued nature of the complex AGM plays an important role, and modular forms make a cameo appearance. Joint work with Thotsaphon Thongjunthug.
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Bilinear forms with the transformation laws A->BAB^T manifest rich Poisson and quantum algebraic structures and admit a number of Poisson reductions among which are reductions to algebras of geodesic functions on Riemann surfaces. We describe the algebroid of bilinear forms, its reductions, the associated braid-group action, generalization to the affine case (joint papers with M.Mazzocco at Advances Math. and Comm.Math.Phys.) and present some new results on possible groupoid structures consistent with the transformation laws.
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I will discuss elliptic curves from the classical number theoretic point of view of trying to solve Diophantine equations. The aim will be both to explain how we think about these creatures and to give an overview of what we can (and sometimes can't) prove about them, and to illustrate it with explicit examples. I will not try to describe the huge modern technical machine that has been developed to study elliptic curves, so most of the results will come as black boxes.
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We give an overview of the algebraic structures underlying linear boundary problems, both in the vector space setting (abstract boundary problems) and in the integro-differential setting (concrete boundary problems). We outline the construction of the polynomial and free objects in the category of integro-differential algebras. The associated ring of integro-differential operators is compared to a generalized Weyl algebra for one variable. We conclude with some remarks on partial integro-differential operators with linear substitutions.
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One of the most important classes of probability measures are those that are infinitely divisible. In Euclidean space, these are characterised by means of their Fourier transform, and this leads to the celebrated Levy-Khintchine formula. In symmetric spaces, there is an analogous characterisation due to Ramesh Gangolli, where the Fourier transform is replaced by the spherical transform, but it only works for measures that are bi-invariant under the action of that subgroup of the isometry group that leaves some point fixed. The problem of obtaining a Levy-Khintchine formula in the general case, without any bi-invariance assumption, has recently been solved in joint work with Tony Dooley (UNSW, now Bath). A key step was finding out what to use instead of the spherical transform. This involves an extension of the concept of a spherical function.
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Test ideals first appeared in the theory of tight closure, and reflect the singularities of a ring of positive characteristic. Motivated by their close connection to multiplier ideals in characteristic zero, N. Hara and K. Yoshida defined generalized test ideals as their characteristic p analogue. Whereas multiplier ideals are defined geometrically, using log resolutions, or even analytically, using integration, test ideals are defined algebraically using the Frobenius morphism. The generalized test ideals of an ideal I form a non-increasing, right continuous family, {τ(c . I)}, parametrized by a positive real parameter c. The points of discontinuity in this parametrization, are called F-thresholds of I and form a discrete subset of the rational numbers (work of Blickle-Mustaţă-Smith, Hara, Takagi-Takahashi, Schwede-Takagi, Katzman-Lyubeznik-Zhang). We consider ideals generated by maximal minors of a matrix of indeterminates, in its polynomial ring over a field of positive characteristic. Using an algebraic approach, we give a complete description of their F-thresholds and generalized test ideals.
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Lie algebras and tangent bundles are the corner cases of Lie algebroids. The notion of ideal in a Lie algebra is well-known, and the Bott connection associated to an involutive distribution can be seen as an ideal in the tangent bundle of a manifold. It is hence natural to ask for the definition of an ideal in a more general Lie algebroid. I will explain why the "old" definitions are not satisfactory. An ideal in a Lie algebra corresponds to a multiplicative distribution on the corresponding Lie group. We describe the counterpart in this spirit of multiplicative distributions on Lie groupoids. These have drawn some attention in connection to geometric quantization of Poisson manifolds, and in a modern approach to Cartan's work on Lie pseudogroups. We explain how the infinitesimal counterpart of multiplicative distributions gives rise to a new notion of ideal in a Lie algebroid.
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The quadratic reciprocity law is a special kind of theorem. It is simple to state but difficult to explain, and it points towards some important hidden structures in arithmetic. I'll talk about a new perspective on quadratic reciprocity, which lets one "see" why it holds as a consequence of some facts concerning the topology of tori. This topological perspective has applications beyond the quadratic reciprocity law. I'll also touch on some of these, namely: a new proof of the Artin reciprocity law, an explanation for Deligne's recent observations on the theory of half-integral weight modular forms (this represents work in progress), and, from the side of homotopy theory, an adelic perspective on the classical "image of J" elements of the stable homotopy groups of spheres.
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The Khovanov homology of a knot categorifies the Jones polynomial and does a whole bunch of things the Jones polynomial can’t. So it’s a sexy thing. In this talk we will remember how Khovanov works and then reinterpret it via sheaf theory. This then allows us to bring in ideas from (stable) homotopy theory. This is joint work with Paul Turner (Geneva).
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The Witten-Dijkgraaf-Verlinde-Verlinde equations first appear in the early '90s in two distinct settings: Topological Quantum Field Theory (where Gromov-Witten invariants are encoded with the structure of a field theory) and Seiberg-Witten theory (a low-energy effective field theory). Links with other areas of mathematics developed rapidly - for example, with singularity theory and integrable systems. The talk will review the three most elementary classes of solutions: rational, trigonometric and elliptic solutions, and discuss ideas of almost-duality and modular Frobenius manifolds.
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We will review some longstanding open questions from noncommutative algebra, namely:
* Is a finitely generated algebraic algebra which is a domain finite dimensional? (The Kurosh conjecture on domains)
* Does a finitely generated ring which is infinite and which is also a division ring exist? (Latyshev)
* The Koethe conjecture on nil rings
* The Jacobson conjecture on finitely presented algebras
We will also look at the following, more recent questions:
* Does a domain exist with a finite but non-integer Gelfand-Kirillov dimension?
* Do finitely generated Noetherian algebras need to have polynomial growth?
We will also consider some connections with other areas, such as group theory and noncommutative algebraic geometry, and look at some methods which are used in this area, for example the Golod-Shafarevich theorem, as well as some partial results which are known to be related to these questions.
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How big is a geometric object? Of course, this question is ill-posed, with many possible answers. In this talk I'll discuss a notion of size, named "magnitude", introduced by Tom Leinster and Simon Willerton. As we'll see, magnitude, which was inspired by category theory, turns out to be related to a surprising array of fields, including integral geometry, potential theory, and even theoretical ecology.
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The moduli space $\bar{M}_{0,n}$ of stable genus zero curves with n marked points is a beautiful space that has been intensively studied by algebraic geometers and topologists for over half a century. It arises from a simple geometric question ("How can we arrange n points on a sphere?"), but is the first nontrivial case of several interesting families of varieties (higher genus curves, stable maps, ...) and phenomena. Despite the long history there are still many mysteries about this variety. I will introduce this moduli space, and highlight some of the recent progress in understanding its geometry.
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Hyperbolic 3-manifolds have been studied intensely by topologists since the mid-1970's. When the fundamental group arises from a certain number theoretic construction (in this case, the manifold is called "arithmetic"), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic hyperbolic 3-manifolds have been studied by number theorists (perhaps not as intensely as the topologists) with different motivations. Very recently, number theorists have started to study the torsion in the homology of arithmetic hyperbolic 3-manifolds. The aim of the first half of this introductory talk, where we will touch upon notions like "arithmeticity", "Hecke operators", will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated "Cheeger-Mueller Theorem" from global analysis.
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The aim of the Minimal Model Program is to generalize the classification of complex projective surfaces, known in the early 20th century, to higher dimensional varieties. Besides providing a historical introduction, we will discuss some recent results and new aspects of this Program.
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Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra $t_n$, associated to A-type hyperplane arrangement. It turns out that Gaudin subalgebras form a smooth algebraic variety isomorphic to the Deligne-Mumford moduli space $\bar M_{0,n+1}$ of stable genus zero curves with n+1 marked points. A real version of this result allows to describe the moduli space of the separation coordinates on the unit sphere in terms of the geometry of Stasheff polytope. A generalisation to Coxeter arrangements will also be discussed. The talk is based on joint work with L. Aguirre and G. Felder and with K. Schoebel.
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I will discuss some recent work that associates and uses noncommutative structures to improve our understanding of algebraic geometry, specifically aspects of the three dimensional minimal model program. The majority of the talk will explain the motivating geometric problems, with lots of pictures, and I will try to motivate and explain why the new noncommutative structures are strictly necessary and should be expected. If time allows, I will discuss a homological method to jump between minimal models, and the newer applications this has to both birational geometry and derived categories.
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One should not expect to be able to say much about the group of automorphisms of an object without first knowing what that object is. But one can produce some automorphisms of cartesian powers of any object using merely the product structure. This is part of a general theory of monoidal categories of a certain form which are governed by special kinds of operads. Both of these structures boil down to a sequence of groups with additional prescribed maps between them, and families like the symmetric groups or braid groups are examples. I will discuss the general theory, some well-known and less-well-known examples, and then try to explain how this sheds light on invertible objects in these monoidal categories. Some of the work in this talk was done in conjunction with Alex Corner, and some of it is still in progress with Ed Prior.
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This will be a gentle introduction to some elements of "non-commutative topology": in particular, using C*-algebras to study groups, or using groups to build interesting examples of C*-algebras, according to taste. I aim to introduce Gelfand duality, motivating how C*-algebras can be thought of as "non-commutative locally compact spaces". I'll then talk about locally compact groups, some functional analysis, and how to build various algebras, aiming to get to discuss the Fourier algebra. Time allowing, I hope to say a little about the theory of locally compact quantum groups.
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The classical Apollonian circle packing is constructed by taking four mutually tangent circles in the unit plane and repeatly inscribing circles in the spaces between them. This results in infinitely many circles, whose radii tend to zero. There is a remarkably simple asymptotic formula due to Kontorich and Oh for the number of these circles with radii > r, as r tends to infinity. This talk will describe this and related results.
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Symplectic manifolds arise naturally in dynamical systems, in Lie theory, and in algebraic geometry. Their study was revolutionised 30 years ago when Gromov and Floer introduced tools from holomorphic curve theory and partial differential equations into the subject; 20 years ago, the subject was revolutionised again by ideas of Fukaya and Kontsevich which suggested that the invariants arising from PDE should be packaged and manipulated via the algebra of A-infinity categories. This talk will give an overview of some of the current progress and problems in the theory.
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This talk concerns nxn matrices with entries from a certain 'tropical' semiring (the entries will be real numbers, however the addition and multiplication operations on this set are not the usual ones). One can consider subsets of R^n which are spanned (under our new operations) by the rows (or columns) of a given tropical matrix; such row and column spaces are called tropical polytopes. An exact characterisation of those tropical polytopes which arise as the column (or row) space of a (multiplicatively) idempotent tropical matrix can be given (joint work with Kambites and Izhakian). A particularly interesting class of tropical idempotents arises from finite (semi)-metric spaces. We garner some further information about the geometric structure of the corresponding tropical polytopes in these cases.
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Simplicial topology is an effective theory for approximating topological spaces by spaces built up out of simplices, and since its early development the theory has played a dominant role in algebraic topology and its applications. In this lecture, I will try to sketch how a similar theory can be developed in which one replaces simplices by tree-like objects. This new theory strictly contains the older simplicial one, and forms an effective tool for the approximation of operads (mathematical objects used to encode algebraic structures on spaces) and of infinite loop spaces.
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For an integer t, t-core partitions are a subclass of partitions that appear naturally in representation theory, number theory, and geometry. More recently, in connection to rational Catalan combinatorics there has been active study into partitions that are simultaneously a-core and b-core, for a, b relatively prime. After a gentle introduction to core partitions, we will explain our recent work connecting simultaneous core partitions with the geometry of lattice points, that in particular allows us to prove a conjecture of Armstrong about the average size of simultaneous core partitions.
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When studying any class of rings or algebras, the existence of a grading often has a big impact on what can be said about the members of the class. In the few years since their inception, cluster algebras have been found in numerous places and have been shown to be responsible for a plethora of combinatorial patterns, but until very recently gradings on cluster algebras have not been considered in a systematic way. In this talk, we will introduce gradings on cluster algebras and show how the intricate structure and combinatorics associated to cluster algebras allows us to find and classify gradings. We will look at cluster algebras of finite type and examine the gradings they admit, making use of cluster categories. Conversely, the gradings bring out some beautiful combinatorics of their own, in the form of tropical frieze patterns.
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Analyzing several classical tests for convergence/divergence of number series, we relax the monotonicity assumption for the sequence of terms of the series. We verify the sharpness of the obtained results on corresponding classes of sequences and functions.
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This talk is about Cartan subalgebras in C*-algebras, and continuous orbit equivalence for topological dynamical systems. These two notions build bridges between operator algebras, topological dynamics, and geometric group theory. We explore rigidity phenomena for continuous orbit equivalence, and discuss how Cartan subalgebras help to understand C*-algebras attached to semigroups of number-theoretic origin.
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The talk will be aimed at a general mathematical audience, not necessarily number theorists.
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The idea of using invariant theory to study arithmetic goes back to Gauss, who studied class groups of imaginary quadratic fields using binary quadratic forms. In recent years, Bhargava and others have revived this circle of ideas, proving striking theorems about the sets of rational points of elliptic and hyperelliptic curves. We will explain some of these ideas.
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The Baum-Connes conjecture asserts that two invariants of a group are isomorphic: the first is a geometric invariant, the K-homology of the classifying space; the second is an analytic invariant, the K-theory of the group C^*-algebra. In some (rare) examples the second invariant can also be viewed geometrically. In this talk I will show that even when the group C^*-algebra can be viewed geometrically, its geometry may be different to that of the classifying space. In the case of (extended) affine Weyl groups the geometries are linked by Langlands duality and the Baum-Connes assembly map can therefore be viewed as a form of Langlands duality.
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Recently a concept known as "interference alignment" has been proposed to increase the transmission capabilities of various wireless networks and ideas from Diophantine approximation, a branch of number theory, are playing a key role. In this talk I hope to give a brief taste of this surprising ,and potentially exciting, link between two seemingly disparate areas.
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There is a lot of interest regarding the growth of invariants of chains of finite index subgroups, e.g. the growth of Betti numbers, rank, deficiency and so on. In this talk I will consider the growth of another invariant: the size of the torsion subgroup in homology. I will focus on two main classes of groups where there has been recent progress: amenable groups (joint with Kar and Kropholler) and right angled groups (joint work with Abert and Gelander). The main tools are from combinatorial group theory and the notion of combinatorial cost.
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The story that led to this talk started with an attempt to understand how to do category theory using a computer. Proofs in category theory can often be represented as commutative diagrams, but how does one describe a commutative diagram in a brief, unique, canonical and helpful fashion? One simple idea turned out to be surprisingly fruitful: not only does it give a language for commutative diagrams, but it led to some new algorithms for a range of computational combinatorial enumeration problems, resulting in contributions to the Online Encyclopedia of Integer Sequences.
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Geometric techniques and results can sometimes shed light on arithmetic problems. I will motivate this statement with a few examples: three elementary puzzles in number theory leading to the question of existence of integral or rational solutions to systems of polynomial equations. The space of complex solutions of these systems are related to surfaces of a very special kind, called K3 surfaces.
I will show how common geometric features of these K3 surfaces can be used to obtain insight on the initial problems.
The talk will be almost entirely self-contained.
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Today there are some very useful notions of a cohomological dimension of a group. Classically, for a torsion-free group, the ordinary cohomological dimension is equal to its geometric dimension provided the cohomological dimension is not two. For groups that contain torsion, the analogues of algebraic and geometric dimensions are less clear. This prompted K.S.Brown's question in 1977 which subsequently led to developments of new notions of cohomological dimensions. I will discuss the history of this topic and a recent joint work Ian Leary on Brown's question.
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An approximate group is a subset S of a group that is almost closed under the group operation (say multiplication), in the sense that the set of pairwise products of elements in S is not much larger than S itself. Evidently a true subgroup forms an approximate group, and much effort has gone into investigating to what extent sets satisfying the aforementioned relaxed closure condition resemble actual subgroups.
The study of approximate groups in the abelian case goes back to the 1970s, but a strongly quantitative as well as a rich non-abelian theory have recently become available, the latter having applications to many other areas of mathematics where groups play a vital role.
This talk will be entirely self-contained. We shall start from the abelian basics before surveying some of the more recent developments, illustrating the richness of the subject and highlighting some of the remaining challenges.
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Two natural numerical invariants that can be associated to a Riemannian manifold are the bottom of the spectrum of the Laplacian operator and, if the manifold are negatively curved, the exponential growth rate of closed geodesics. Suppose we have a regular cover of a compact manifold. Then, for each of these quantities, we might ask under what circumstances we have equality between the number associated to the cover and the number associated to the base. This question becomes non-trivial questions once the cover is infinite. It turns out that the question has a common answer in the two cases and this depends only on the covering group as an abstract group. For the Laplacian, this result was obtained by Robert Brooks in the 1980s, and Rhiannon Dougall and I have recently obtained the analogue for the growth of closed geodesics. I will discuss this work, relating it to random walks and a class of groups introduced by von Neumann in his study of the Banach-Tarski Paradox.
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It is a well known fact that the braid orbits on Nielsen tuples are in one to one correspondence with the connected components of Hurwitz spaces corresponding to curve covers of the Riemann sphere. The theorem of Conway and Parker asserts that if every element of the Schur multiplier of a group G is a commutator and if the ramification type is sufficiently general, then the corresponding Hurwitz space is connected. For many reasons it would be desirable to have effective versions of this theorem. In 2010 Fried showed that the lift invariant distinguishes components of Hurwitz spaces corresponding to alternating group covers whose ramification data consists entirely of three cycles.In particular the number of connected components of the corresponding Hurwitz space is never more than 2. In joint work with A. James and S.Shpectorov we show that the lift invariant distinguishes Hurwitz components of $A_5$ covers. Generalizations of this will also be discussed.
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Given a finite group G and subgroups G_1,...,G_n, for S a subset of {1,...,n} let h_S be the log of the index of the intersection of the G_i for i in S and let h = (h_S), a point in 2^n-dimensional real space. A fundamental question is to describe the conic closure of these points as G and its subgroups vary. This set arises in many fields- it has alternative definitions in terms of polymatroids or of joint entropies of discrete random variables. It interests engineers since finding network coding capacities is a convex optimization problem on the set. It is, however, only explicitly known for n=2 and 3. For n=4 its boundary is curved and I will describe work with Ting-Ting Nan that describes a little more about this mysterious region.
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In 1979, D.Mumford constructed a celebrated example of a fake projective plane, but the same paper contains an outline of a much more general construction of "fake" algebraic surfaces using groups acting on buildings. We'll discuss explicit constructions of such groups, old and new, and their connections with the algebraic surfaces.
Main results are based on joint works with N.Boston, N.Barker, N.Peyerimhoff and J.Stix.
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Finite metric graphs paramaterize many phenomena in mathematics and science, so we would like to understand spaces which parameterize such graphs, i.e. moduli spaces of graphs. Moduli space of graphs with a fixed number of loops and leaves often have interesting topology that is not at all well understood. For example, Euler characteristic calculations indicate a huge number of nontrivial homology classes, but only a very few have actually been found. I will discuss the structure of these moduli spaces, including recent progress on the hunt for homology based on joint work with Jim Conant, Allen Hatcher and Martin Kassabov.
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In this talk I will try to explain how one tries to classify algebraic varieties, in a birational sense, using certain special varieties as building blocks. By special varieties I mean Fano's, Calabi-Yau's, and varieties of general type. These varieties are also of great interest in other parts of mathematics such as arithmetic and differential geometry.
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In 2014, Maryam Mirzakhani of Stanford University became the first women to be awarded the Fields medal. The starting point of her work was a remarkable relationship called McShane’s identity, about the lengths of simple closed curves on certain hyperbolic surfaces. The proof of this identity, including the Birman-Series theorem about simple curves on surfaces, uses only quite basic ideas in hyperbolic geometry which I will explain. We will then look briefly at Mirzakhani’s ingenious way of exploiting the identity. Time permitting, we will also touch on some other open questions about McShane’s identity. The talk should be accessible to advanced undergraduates.
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We shall examine whether the story of the Jacobi identity starts with Jacobi, circa 1840, and whether it should rather be called by the name of the British mathematician, William F. Donkin, concluding however that it was Jacobi's. Forty years later, Sophus Lie defined algebras with a skew-symmetric bilinear composition law satisfying the Jacobi identity, which he referred to as "the analytical foundation of my theory of transformations," algebras that were eventually called "Lie algebras" by Hermann Weyl in 1933. The concept of ``Leibniz algebras'', also called "Loday algebras" because of Loday's influential 1993 paper, appeared when the bracket was no longer assumed to be skew-symmetric and the Jacobi identity was suitably written "in Leibniz form". I shall try to give an idea of the many recent variants and generalizations of the Jacobi identity, both in mathematics and in mathematical physics, such as graded Jacobi identities, Jacobi identity for non-commutative double brackets, hom-Jacobi identity, and the Jacobi identity in the theory of triple tangent bundles, of Lie algebroids and of Courant algebroids, or the related properties of classical $r$-matrices and $\mathcal O$-operators.
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One way of understanding infinite groups is by investigating their actions on special spaces, such as Hilbert and Banach spaces, non-positively curved spaces etc. Kazhdan property (T) is formulated in terms of actions on Hilbert spaces, and turns out to be relevant in many different areas. Several strengthened versions of property (T) in the setting of Banach spaces have been formulated in recent years, each of them interesting for different reasons: relevance for the conjectures of Baum-Connes and Novikov, separation between rank one and higher rank, examples of expanders with stronger properties, presumed connection to the conformal dimension of the boundary, stronger rigidity results etc.
In this talk I shall overview various generalisations of property (T) to Banach spaces, especially in connection with random walks, expanders and random graphs and groups. This is based on joint work with J. Mackay and P. Nowak.
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The talk concerns a construction, due to the physicist Gaiotto, of complex Lagrangian submanifolds in the moduli space of Higgs bundles over a Riemann surface. These are BAA-branes in the physicists’ terminology. Looking at particular examples reveals different aspects of the geometry of Higgs bundle moduli spaces, with links to the classical geometry of the Kummer surface.
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Mumford's geometric invariant theory (GIT) developed in the 1960s provided a method for constructing quotient varieties for linear actions of reductive groups on affine and projective varieties, and has many applications (for example in the construction of moduli spaces in algebraic geometry). The aim of this talk is to discuss an extension of Mumford's GIT to actions of linear algebraic groups which are not necessarily reductive, and some of its applications.
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There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other L-functions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices. Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into long-standing questions in number theory. I will give a history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant (work with Brian Conrey and Amy Mason).
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In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.
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Some natural problems in computational algebra and cryptography require analysis of finite algebraic structures (for example, groups or fields) up to homomorphisms computable by randomised algorithms working in probabilistic polynomial time. This opens a fascinating new chapter of algebra, where we need to construct, by randomised algorithms, some structures (for example, fields) within structures of different nature (for example, groups).
I will explain, as a na example, an algorithm that solves a key problem in probabilistic computational group theory. The problem was formulated in 1999 by Babai and Beals but remained intractable since then. The question a simple: you are given a few n by n matrices over a finite field of characteristic p > 0. You know that they generate a subgroup X isomorphic to SL_2(p^k). Find, in this group X, an element of order p. The catch is that X can have an astronomical number of elements, while unipotent elements are exceptionally rare. It is a search for a needle not even in the proverbial haystack, ibut in the Universe.
Our algorithm has one unusual aspect. On one hand, it is an efficient practical algorithm that can be run on any laptop; on the other hand, it imitates the history of geometry of the physical world in its development from Euclid to Lobachevsky, Minkowski, Lorentz, and Planck.
(joint work with Sukru Yalcinakaya)
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We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most X^{1-c} elements less than X) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.
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We will be interested in the problem how rational points are distributed and, in particular, in analysing discrepancy of the distribution. We describe approaches for obtaining upper and lower bounds on discrepancy. It turns out that this problem leads to an interesting interplay between arithmetic geometry, ergodic theory, and the theory of automorphic representations. This is a joint work A. Ghosh and A. Nevo.
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Cluster algebras are commutative algebras constructed from quivers by a recursive process called mutation. They were introduced by Fomin and Zelevinsky around 2000 and there are now established connections with many areas of mathematics. Poisson structures were brought into the picture by Gekhtman, Shapiro and Vainshtein in 2003 and the closely related quantum cluster algebras followed in a 2005 paper by Berenstein and Zelevinsky. Quantum cluster algebras are fascinating examples of noncommutative rings and it is from the point of view of noncommutative ring theory that I will discuss them. There have been several papers where known noncommutative algebras, or classes of such algebras, were shown to have quantum cluster algebra structures. An alternative approach is to look for interesting new noncommutative algebras by determining quantum cluster algebras, and the associated Poisson structures on the corresponding commutative cluster algebras, arising from particular quivers. I will start with a simple example (two vertices and one arrow) to illustrate the ideas and then look at the quantum cluster algebras for a family of quivers appearing in the classification of periodic quiver mutation by Fordy and Marsh. The aims will include to present the quantum cluster algebra by generators and relations and to decide whether the algebra is noetherian. The original content of the talk is joint work with Christopher Fish.
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Let $\pi(x)$ denote the number of primes up to $x$, let $\mathrm{li}(x)$ denote the logarithmic integral. The prime number theorem says that $$ \pi(x) \sim \mathrm{li}(x) $$ i.e. that the ratio tends to $1$ as $x \to \infty$. If you look at any table of primes, you will find that $\pi(x) < \mathrm{li}(x)$. However, Littlewood proved in 1914 that the difference $\pi(x) - \mathrm{li}(x)$ changes sign infinitely often. This means that there is a least $X$ for which $\pi(X) > \mathrm{li}(X)$. What is $X$? No-one knows. Any upper bound for $X$ is a Skewes number. In the course of the 20th century, successive upper bounds were discovered, culminating in the Skewes number $\exp 727. 9513$. This number has $312$ digits. All the proofs depend on the explicit formula for $\pi(x)$ due to Riemann in his memoir of 1859.
This talk will be accessible to staff, graduate students, and undergraduates with a first course in complex analysis.
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I will discuss some of the issues that are thrown up when we try to describe the space of compact group automorphisms modulo various natural equivalences coming from dynamical systems. Much of this is joint work with Stephan Baier (Jawarharlal Nehru University, India), Richard Miles (who is in Sheffield), Shaun Stevens (UEA), Sawian Jaidee (Khon Kaen, Thailand) and Jason Bell (Waterloo, Canada).
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I will report on results obtained jointly with R.Marsh. We construct the superpotential of a Grassmannian X as a regular function W on the complement of the anticanonical divisor on a Langlands dual Grassmannian, X^, and prove that it encodes Gromov-Witten invariants of the original Grassmannian via an associated Gauss-Manin system.
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The motivation behind this talk is to make precise the feeling that if an analytic (or otherwise well-behaved) function takes, in a suitable sense, many rational values for rational arguments, then there should be algebraic reasons for this. For example, a result of Bombieri and Pila from 1989 states that if $f:[0, 1] \to [0, 1]$ is a real analytic function, then either $f$ is algebraic (over $\mathbb{Q}(x)$) or else for all $\epsilon > 0$ and all sufficiently large $H$, there are at most $H^{\epsilon}$ pairs of rationals $p, q$ with denominators bounded by $H$ such that $f(p) = q$. It was suspected at the time that a similar result should be true for functions of many variables and much work was done in dimensions two and three by using the classical theory of analytic sets. However, the general case was finally solved (by Pila and myself in 2006) by setting the problem in the much wider and more flexible framework of o-minimal structures (a notion from Mathematical Logic). I shall discuss these developments and, if time permits, their applications to some classical number theoretic problems.
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Poisson structures were originally invented in the early 19th century to provide a framework for optics and classical mechanics. In this talk, we will discuss the existence of symplectic realizations of Poisson manifolds, a longstanding problem which can be traced back to Sophus Lie. In particular, we will present an explicit existence result for arbitrary holomorphic Poisson manifolds.
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Derived commutative rings are supposed to be commutative ring-like entities with additional homotopy theoretic data. There are several candidates for these including simplicial commutative rings, E-infinity dgas, E-infinity ring spaces or spectra or equivalent topology versions.
I will explain the basic ideas involved in introducing homotopical versions of algebraic structures, then discuss the implications for the homology of spaces with such structure.|
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The conjecture of Birch and Swinnerton-Dyer is one of the principal open problems in number theory today. In my lecture, I shall give a brief account of the history of the conjecture, its precise formulation, and the partial results obtained so far in support of it.
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Standard commutative algebra is based on commutative monoids, Abelian groups and commutative rings. In recent years, there has been some progress in developing an area that may be referred to as commutative 2-algebra, in which the familiar notions used in commutative algebra are replaced by their categorified counterparts (for example, commutative monoids are replaced by symmetric monoidal categories). The aim of this talk is to explain the analogy between standard commutative algebra and commutative 2-algebra, and to outline how this analogy suggests analogues of basic aspects of algebraic geometry. In particular, I will describe how some joint work with Andre’ Joyal on operads and analytic functors fits in this context.
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In 2009, three Japanese theoretical particle physicists observed that the elliptic genus of a K3 surface, when expressed in terms of mock modular forms, exposes numbers that can be linked to the dimensions of finite dimensional representations of the sporadic group Mathieu 24.
Since then, this intriguing connection has been studied from several points of view, other examples of the same type of phenomenon for other finite groups and mock modular forms have been discovered, and the research topic of `New Moonshines’ has slowly caught the attention of researchers across fields.
In this talk, I will describe the 2009 observation, now referred to as `Mathieu Moonshine’, and explain the challenges faced by the theoretical physics community in understanding the origin and role of the huge Mathieu 24 finite symmetry in the context of strings compactified on K3 surfaces. In particular, I will discuss how this phenomenon is related to the geometry of K3 surfaces and introduce the concept of symmetry surfing.
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Let $G$ be a group, $K$ a field and $A$ a $n$ by $m$ matrix over the group ring $K[G]$. Let $G=G_1>G_2>G_3\cdots$ be a chain of normal subgroups of $G$ of finite index with trivial intersection. The multiplication on the right side by $A$ induces linear maps $$\begin{array}{cccc} \phi_i: & K[G/G_i]^n & \to& K[G/G_i]^m\\ &&&\\ &(v_1,\ldots,v_n) &\mapsto& (v_1,\ldots,v_n)A.\end{array}$$ We are interested in properties of the sequence $\{\frac{\dim_K \ker \phi_i}{|G:G_i|}\}$. In particular, we would like to answer the following questions.
- Is there the limit $ \lim_{i\to \infty}\frac{\dim_K \ker \phi_i}{|G:G_i|}$?
- If the limit exists, how does it depend on the chain $\{G_i\}$?
- What is the range of possible values for $ \lim_{i\to \infty}\frac{\dim_K \ker \phi_i}{|G:G_i|}$ for a given group $G$?
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The tube formulas of Steiner and Weyl express the measure of tubular neighbourhoods of geometric objects (convex sets and Riemannian manifolds, respectively) as polynomials with certain curvature invariants as coefficients. We introduce these formulas and discuss recent applications to fields such as geometric probability, concentration of measure, numerical analysis, and convex optimization. Based on work with D. Amelunxen, M.B. McCoy, J.A. Tropp, F. Cucker, P. Buergisser
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Riemann's moduli spaces are at the heart of much modern mathematics. In this lecture we will explore their properties as an operad. Operads were introduced in the 1970 in homotopy theory to study loop spaces. Infinite loop spaces are of particular interest as they give rise to generalised cohomology theories. In the 1990's operads had a renaissance with much interest stimulated from mathematical physics. In particular, Segal's axiomatic approach to conformal field theory defines an operad of Riemann surfaces. We will show that this is an example of a new generation of operads detecting infinite loop spaces. The talk will introduce the main concepts and is addressed to a general mathematical audience.
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Every finite regular CW complex is, ipso facto, a cohomologically stratified space when filtered by skeleta. In this talk, I will outline a method to discover the canonical (i.e., coarsest possible) stratification of such a complex that is compatible with its underlying cell structure. The construction proceeds by first localizing and then resolving a complex of cosheaves which capture local cohomology at every cell. The result is a sequence of categories whose limit recovers the desired strata via its (isomorphism classes of) objects. As a bonus, the entire process is algorithmic and amenable to efficient computations.
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In 1962, Hartshorne proved that the dual graphs of an arithmetically Cohen-Macaulay scheme is connected. After establishing a correspondence between the languages of algebraic geometry, commutative algebra and combinatorics, we are going to refine Hartshorne's result and measure the connectedness of the dual graphs of certain projective schemes in terms of an algebro-geometric invariant of the projective schemes themselves, namely their Castelnuovo-Mumford regularity. Time permitting, we are also going to address briefly the inverse problem of Hartshorne's result, by showing that any connected graph is the dual graph of a projective curve with nice geometric properties. This is joint work with Bruno Benedetti and Matteo Varbaro.
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A continuous map $\mathbb{R}^m \rightarrow \mathbb{R}^N$ or $\mathbb{C}^m \rightarrow \mathbb{C}^N$ is called $k$-regular if the images of any $k$ distinct points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of $N$ for which such maps exist. The methods of algebraic topology provide lower bounds for $N$, however there are very few results on the existence of such maps for particular values m. During the talk, using the methods of algebraic geometry, we will construct $k$-regular maps. We will relate the upper bounds on the minimal value of $N$ with the dimension of the a Hilbert scheme. The computation of the dimension of this space is explicit for $k< 10$, and we provide explicit examples for $k$ at most $5$. We will also provide upper bounds for arbitrary m and k. The problem has its interpretation in terms of interpolation theory: for a topological space X and a vector space $V$, a map $X \rightarrow V$ is k-regular if and only if the dual space $V^*$ embedded in space of continuous maps from $X$ to the base field $\mathbb{R}$ or $\mathbb{C}$ is $k$-interpolating, i.e. for any $k$ distinct points $x_1,...,x_k$ of $X$ and any values $f_i$, there is a function in $V^*$, which takes values $f_i$ at $x_i$. Similarly, we can interpolate vector valued continuous functions, and analogous methods provide interesting results.
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In this talk I will give an overview of mathematical structures used in modern quantum filed theory. I will focus on notions from functional analysis, like nets of operator algebras, and show how these combine with homological algebra methods to provide a rigorous description of perturbative gauge theories on curved spacetimes and of effective quantum gravity. The framework I present is called perturbative algebraic quantum field theory (pAQFT) and it is an emerging new way of approaching mathematical foundations of QFT.
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Mutations of quivers were introduced by Fomin and Zelevinsky in 2002 in the context of cluster algebras. For some classes of quivers, mutations can be realised using geometric or combinatorial models. We will discuss a construction of a geometric model for all acyclic quivers. The construction is based on the geometry of reflection groups acting in quadratic spaces. As an application, we show an easy and explicit way to characterise real Schur roots (i.e. dimension vectors of indecomposable rigid representations of Q over the path algebra kQ), which proves a recent conjecture of K.-H. Lee and K. Lee for a large class of acyclic quivers.
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One of the motivating problems in ring theory in the past twenty-five years has been the classification of noncommutative projective surfaces: that is, classifying all noetherian N-graded rings of cubic growth. In particular, one may ask: Fix a division ring D. What are the N-graded rings as above that are contained in the polynomial extension D[t] and have the same (graded) division ring of fractions? This is known as "classifying noncommutative surfaces birational to D''.
This question is particularly interesting where D is the division ring which comes from the famous Sklyanin algebra: a graded ring which behaves like the coordinate ring of a noncommutative version of the projective plane. Remarkably, although this situation is highly noncommutative, many of the famous theorems of (commutative) algebraic geometry of surfaces have very strong analogues. We describe how to do birational geometry in this noncommutative context, including noncommutative versions of blowing up a point and contracting a curve. However, these techniques, when applied to noncommutative rings, have applications which are extremely counterintuitive when compared with the commutative context.
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We show an application of the new notion of a thick morphism of (super)manifolds.
For an arbitrary manifold $M$, consider the supermanifolds $\Pi TM$ and $\Pi T^*M$, where $\Pi$ is the parity reversion functor. The space $\Pi TM$ has an odd vector field that can be identified with the canonical de Rham differential $d$; functions on it can be identified with differential forms on $M$. The space $\Pi T^*M$ has an odd Poisson bracket $[ - , - ]$; functions on it can be identified with multivector fields on $M$ and the bracket is the canonical Schouten bracket. An arbitrary even function $P$ which obeys the master-equation $[P,P]=0$ defines an even homotopy Poisson structure on the manifold $M$ and an odd homotopy Poisson structure (the "higher Koszul brackets") on differential forms on $M$.
In the case when the function $P$ is quadratic on fibres, then the homotopy Poisson structure on $M$ and the higher Koszul bracket on differential forms are ordinary even and odd Poisson structures. It is a classical fact that there is a linear map of differential forms endowed with the Koszul bracket to multivector fields endowed with the canonical odd Schouten bracket $[ -, - ]$. In the general case, when we have a homotopy Poisson structure on $M$, this linear map does not exist. We show how to construct a non-linear transformation from differential forms endowed with the higher Koszul brackets to multivector fields with the canonical Schouten bracket. This is done as a non-linear pullback with respect to some thick morphism of supermanifolds, a notion recently introduced.
(The talk is based on the work with Ted Voronov.)
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Finite groups acting linearly on complex projective spaces have been studies by many people including Blichfeldt, Brauer, Lindsey, Wales, Collins, Thompson and Robinson. In dimension one (projective line) they had been classified in antiquity. Aside from cyclic and dyhedral groups, there are just three such groups, which are the groups of symmetries of Platonic solids. In higher dimensions, the classification is much more complicated. Finite subgroups of the projective transformations of the plane have been classified by Blichfeldt in 1917. He also classified finite subgroups of projective transformations of the three-dimensional space. In my talk I will describe Blichfeldt's classification and explain how to use it to describe equivariant birational geometry of the projective plane and three-dimensional space.
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A conjecture of Juan-Pineda and Leary states that any group admitting a cocompact model for the classifying space for the family of virtually cyclic subgroups has to b be virtually cyclic already. This conjecture has been proved for large classes of groups.
In this talk I will give an overview of some of these results and constructions, will discuss a weakened condition for these spaces, and will give examples of groups satisfying this condition. This is joint work with N. Petrosyan.
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The notion of Frobenius structures was introduced by Dubrovin in the '90s as a geometric axiomatization of 2 dimensional Topological Field Theories. A Frobenius manifold is essentially a manifold whose tangent spaces at any point are endowed with the structure of associative Frobenius algebra, varying "smoothly" with respect to a metric. The associativity is encoded in a system of non-linear PDE called WDVV equations.
The talk is an introduction to Frobenius manifolds and to their link with the theory of isomonodromic deformations and with WDVV equations.
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Random convex polytopes have been extensively studied over the last decades. A popular model of such polytope is the convex hull of a random walk (which is a random sequence whose increments are independent random vectors with identical distribution). I will present a problem on such convex hulls with notable connections to conic geometry and combinatorics. This is a joint work with Zakhar Kabluchko (Munster) and Dmitry Zaporozhets (St. Petersburg).
Consider the probability that the convex hull of an n-step random walk in R^d does not absorb the origin, which in dimension one means that the trajectory of the walk does not change its sign. The remarkable formula of Sparre Andersen (1949) states that any one-dimensional random walk with symmetric continuous distribution of increments stays positive with probability (2n-1)!!/(2n)!!, which does not depend on the distribution. We prove a multidimensional distribution-free counterpart of this result and give an explicit tractable formula for the absorption probability. Our idea is to show that the absorption problem is equivalent to a geometric problem on counting the number of Weyl chambers of type B_n in R^n intersected by a generic linear subspace of co-dimension d. As the main application of this result, we obtain explicit distribution-free formulas for the expected number of faces and vertices of the convex hulls of the random walk.
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In topology, the existence of a nowhere vanishing section of an oriented vector bundle is detected by its Euler class (in case rank of vector bundle equals dimension of base). This is classical and goes back to at least the 1950s. The analogous story in algebra is less classical and has led to deep questions and results in algebraic K-theory, algebraic cycles, A1-homotopy and group homology. After recalling the topological story, I will give a survey of the algebraic side.
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Hilbert modular forms were introduced by David Hilbert in 1892 in an attempt to generalise so called elliptic modular forms to other settings. Considered to be a notoriously difficult topic, it wasn't until the mid 1970s that they were seriously studied, notably by Goro Shimura. Since then, they have become very central objects to modern number theory.
In this talk, we will start with a gentle introduction to Hilbert modular forms. Then, we will discuss various applications to number theory and arithmetic geometry.
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A crucial result in Shimura's work on the special values of L-functions of modular forms concerns the existence of a twisting character to ensure that a twisted L-value is nonzero at the center of symmetry. Even for simple situations involving L-functions of higher degree this problem is open: for example, if $\pi$ is the automorphic representation attached to a holomorphic cusp form, then it has been an open problem to find a character such that the twisted symmetric cube L-function of $\pi$ does not vanish at the center.
We will present a recent joint work with F. Januszewski and A. Raghuram in which purely arithmetic methods involving studying p-adic distributions on Galois groups are used to tackle this problem.
Given a cohomological unitary cuspidal automorphic representation $\Pi$ on GL(2n) over a totally real field, under a very mild regularity assumption on the infinity type that ensures two critical points for the standard L-function of $\Pi$, supposing $\Pi$ admits a Shalika model, then for any ordinary prime p for $\Pi$, we prove that for all but finitely many Hecke characters the twisted central L-value of $\Pi$ does not vanish.
For example, with a classical normalization of $L$-functions, it follows from our results that there are infinitely many Dirichlet characters $\chi$ such that $L(6, \Delta \otimes \chi) L(17, {\rm Sym}^3\Delta \otimes \chi) \neq 0$ for the Ramanujan $\Delta$-function.
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I will talk about the following problem in algebraic geometry: given a family of algebraic varieities, if general fibers are rational, are all fibers rational? The talk will be based on recent joint work of myself and Nicaise, and a development by Kontsevich and Tschinkel.
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I will report on a new connection between cluster algebras and continued fractions given in terms of so-called 'Snake graphs'. Snake graphs are planar graphs first appeared in the context of cluster algebras associated to marked surfaces. In their first incarnation, they were used to give formulas for generators of cluster algebras. Along with further investigations and several applications, snake graphs were also studied from a more abstract point of view as combinatorial objects. This talk will focus on a combinatorial realisation of continued fractions in terms of 'perfect matchings' of snake graphs. I will also discuss applications to Cluster algebras, to elementary Number theory and, time permitting, to Knot theory. This is joint work with Ralf Schiffler.
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Ever since Riemann's seminal paper in 1859, analytic approaches to number theory have developed out of an understanding of the zeros of the Riemann zeta-function. In 2009, Soundararajan and I proposed an alternative approach (the "pretentious approach"), piecing together many "ad hoc" ideas from the past into a coherent theory. This new theory has taken on a life of its own in the last few years, providing the framework for some impressive new results by Matomaki, Radziwill, Tao and others. In this talk we will explain the main ideas and try to give some sense of how these new works fit in.
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The Baum-Connes conjecture is the "commutative" part of Alain Connes' noncommutative geometry programme, since it forms the bridge to classical geometry and topology. In its classical form, the conjecture identifies two object associated with a countable discrete group: one analytical and one topological.
In their 1997 paper, Davis and Lück utilised a (then little known) category theoretic variant of an operator algebra to present a unified approach to this conjecture, and to the Isomorphism Conjecture of Farrell and Jones on the algebraic K- and L-theory of integral group rings.
In this talk we will look at how this machinery fits together, what information the machinery gives us about the identifying maps, and how we might go about extending the scope of the techniques to topological groups and groupoids.
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Miles Reid proved that in characteristic zero, a general hyperplane section of a canonical (resp. klt) 3-hold has only rational double points (resp. klt singularities). His proof heavily depends on the Bertini theorem for free linear series, which fails in positive characteristic. Thus, it is natural to ask whether the same statement holds in positive characteristic or not. In this talk, I will present an affirmative answer to this question when the characteristic is larger than 5. This is joint work with Kenta Sato.
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The local Langlands correspondence is a web of sometimes conjectural correspondences between, on the one hand, irreducible representations of reductive groups over a p-adic field F and, on the other hand, certain representations of the absolute Weil group of F (which is almost the absolute Galois group). I will try to explain what the objects involved here are, some of what the correspondence predicts and what is known/unknown, as well as work (particularly due to Bushnell and Henniart for general linear groups) towards making the correspondence explicit. Hopefully I will also explain some joint work (with Blondel and Henniart) where we describe the "wild part'' of the correspondence for symplectic groups.
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We consider $C^*$-algebras $A_{q_i}\Theta$ generated by relations of the following form $$a_i^*a_i=1+q_i a_ia_i^* a_i^*a_j=e^{2\pi\theta_{ij}} a_ja_i^*, \quad i\ne j i, \quad j=1,\ldots,d $$ where $-1 \lt q_i \lt 1$, $\theta_{ij}=-\theta_{ji}$, $i\ne j$.
We show that $A_{q_i}\Theta \simeq A_{0}\Theta$ is an extension of higher-dimensional non-commutative tori and study its properties.
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I argue that pure mathematics is walking inexorably towards a cliff edge, and that anyone who believes that current pure mathematics is rigorous, or a science, needs to wake up and look at the facts, which there will be plenty of in this talk, and they are not pretty. Are our results reproducible? Does it matter? What *is* mathematics? Can computer scientists save us? Can *undergraduates* save us? I hope so. This talk is about pure mathematics but will be accessible to undergraduates, mathematicians both pure and applied/applicable and computer scientists.
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A central theme of research in operator algebras is the Baum-Connes conjecture, which predicts the K-theory of group $C^*$-algebras and crossed products. In this talk I will give a leisurely introduction to this conjecture, explain what it is good for, and discuss some recent connections with the theory of quantum groups.
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I'll talk about an ongoing joint work with Gabor Elek about approximation of groups with respect two the rank metric. The basic question is the following variant of the Halmos problem about commuting matrices: if A and B are large matrices such that the rank of the image of the commutator is small, is it true that A and B can be perturbed with small rank matrices in such a way that the resulting matrices commute? There are interesting connections to classical notions of commutative algebra, in particular we develop what are perhaps some new (or forgotten) variants of Nullstellensatz for primary ideals.
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The key ingredient in Wiles' proof of Fermat's last theorem was to establish the modularity of elliptic curves. Despite many impressive advances in the Langlands programme the analogous question of modularity for abelian varieties of dimension 2 is still open. I will discuss what we know and present joint work with Kris Klosin (CUNY) on the modularity of abelian surfaces which have a rational torsion point.
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Mirror symmetry is a broad correspondence between algebraic and symplectic geometry. It is a bit scary in the beginning as a true understanding of it requires some knowledge of both of these rather deep fields. In this talk, I will not give you a true understanding, rather I will provide examples of how fascinating this correspondence is. My intention is to get you "hooked" - it's up to you to decide whether you want to pursue this for a true understanding.
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This talk aims to be a journey through the history of the Newton polygon. To any multivalued polynomial we can associate a convex polytope in Euclidean space by taking the convex hull of its exponent vectors. These polygons are named after Newton who, in the 17th century, made use of them in the setting of infinitesimal calculus. In the late 19th century they were employed by Baker to compute the genus of plane curves. An extensive study of the relation between hypersurfaces (zero loci of polynomial equations) and Newton polytopes took off in the 20th century with the advent of toric algebraic geometry. A further generalisation was introduced by Okounkov, in the last two decades, to study further properties of polarised algebraic varieties.
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Which properties of a group are determined by the set of its finite quotients? We give an introduction to this classical question and present examples and non-examples of such "profinite" properties. Afterwards we take a closer look at profinite properties of arithmetic groups. An arithmetic group is, roughly speaking, a group of matrices with integer entries. We present a property which is surprisingly determined by the finite quotients and we try to explain this phenomenon. In the end, we mention possible generalizations and open problems. This is based on joint work with H. Kammeyer, J. Raimbault and R. Sauer.
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The braid groups were defined by Artin in 1925, and are usually defined in terms of strings in 3-dimensional space. However there is a fruitful 2-dimensional perspective of the braid groups as homeomorphisms (up to some natural equivalence) of a disc with holes, in other words, the braid groups are special cases of mapping class groups of surfaces. Mapping class groups can be viewed in a number of ways, and are of interest in several different fields, such as dynamics, algebraic geometry, surface bundles, hyperbolic geometry, to name a few. A key theorem that demonstrates this intradisciplinary feature is the Nielsen--Thurston classification. I will explain what the Nielsen--Thurston classification is, describe some basic examples and analogies, and highlight its importance. I will then explain how to view this from the geometric group theory perspective, and discuss my work with Mark Bell that uses this point of view to solve the conjugacy problem for mapping class groups in polynomial time. At the end of the talk I will discuss some new ideas that may lead to applications in knot theory via the braid groups.
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Consider a billiard table shaped as a Euclidean polygon with labeled sides. A ball moving around on the table determines a bi-infinite “bounce sequence” by recording the labels of the sides it bounces off. We call the set of all possible bounce sequences the “bounce spectrum” of the table. In this talk I will explain why the bounce spectrum essentially determines the shape of the table: with the exception of a very small family (right-angled tables), if two tables have the same bounce spectrum, then they have to be related by a Euclidean similarity. The main ingredient in proving this fact is a technical result about non-singular geodesics on surfaces equipped with flat cone metrics. This is joint work with Moon Duchin, Chris Leininger, and Chandrika Sadanand.
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In 1916, Ramanujan made a conjecture that can be stated in completely elementary terms: he predicted an upper bound on the coefficients of a power series obtained by expanding a certain infinite product. In this talk, I will discuss a more sophisticated interpretation of this conjecture, via the Fourier coefficients of a highly symmetric function known as a modular form. I will give a hint of the idea in Deligne’s proof of the conjecture in the 1970’s, who related it to the arithmetic geometry of smooth projective varieties over finite fields. Finally, I will discuss generalisations of this conjecture and some recent progress on these using the machinery of the Langlands program. The last part is based on joint work with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne.
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In what sense can automorphic forms or Galois representations be viewed as rational points on an algebraic variety? One way to explore this question is by counting arguments. The first result in this direction dates back to an early theorem of Drinfeld, which computes the number of 2-dimensional Galois representations of a function field in positive characteristic; the resulting expression is reminiscent of a Lefschetz fixed point theorem on a smooth algebraic variety over a finite field. More recently it was observed that in the number field setting there are formal similarities between the asymptotic counting problems for rational points on Fano varieties and for automorphic representations on reductive algebraic groups. Very little is known in the latter context. I’ll discuss joint work on this topic with Djordje Milicevic, in which we (mostly) solve the automorphic counting problem on the general linear group. Our results can be viewed as being analogous to the well-known result of Schanuel on the number of rational points of bounded height on projective spaces. If time permits, I may also present a short argument, using sphere packings in large dimensions, to give upper bounds on such automorphic counts.
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Complex dynamics concerns the iteration of analytic functions of the complex plane. For each function, the plane is split into two sets: the Fatou set (where the behaviour of the iterates is stable under local variation) and the Julia set (where the behaviour is chaotic). The dynamical behaviour of the iterates inside the periodic components of the Fatou set was classified into four different types by the founders of the subject and this classification has played a key role in the development of the subject. One of the most dramatic breakthroughs was given by Sullivan in the 1980s when he proved that, for rational functions, all components of the Fatou set are eventually periodic and there are no so-called wandering domains. For transcendental functions, however, wandering domains can exist and the rich variety of possible behaviours that can occur is only just becoming apparent.
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Theory of divisors on graphs is analogous to the classical theory for algebraic curves. The combinatorial language in this setting is "chip-firing game” which has been independently introduced in other fields. A divisor on a graph is simply a configuration of dollars (integer numbers) on its vertices. In each step of the chip-firing game we are allowed to select a vertex and then lend one dollar to each of its neighbors, or borrow one dollar from each of its neighbors. The goal of the chip-firing game is to get all the vertices out of debt. In this setting, there is a combinatorial analogue of the classical Riemann-Roch theorem. I will explain the mathematical structure arising from this process and how it sits in a more general framework of (graphical) hyperplane arrangements.
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The idea of moduli spaces classifying structures in various fields of mathematics dates back to Riemann who tried to classify complex structures on a compact surface. It took another hundred years and many ingenious ideas of Grothendieck, Mumford and other mathematicians to write down a proper definition of moduli spaces and to construct nontrivial examples including Riemann‘s vague idea of a moduli space of complex structures on a surface. However, it became quite obvious that the concept of moduli spaces/stacks developed in the 60‘s and 70‘s is not sufficient to describe all moduli problems. Another 50 years and a fair amount of homotopy theory was needed to provide a definition of moduli spaces having all required properties. A large class of examples comes from (higher) representation theory. The aim of my talk is to provide a gentle introduction into these new concepts and thereby to show how nicely algebraic geometry, topology and representation theory interact with each other. If time permits, I will also sketch applications in Donaldson-Thomas theory.
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I will introduce the Gross-Stark units and present their application to Hilbert’s 12th problem. Following an earlier work in special case with Darmon, Dasgupta gave precise conjectural p-adic analytic formula for these units. After giving a formulation of this conjecture, I will sketch a proof of this conjecture. This is a joint work in progress with Samit Dasgupta.
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I will discuss a conjecture about stabilisation of deficiency in finite index subgroups of a finitely presented group and relate it to the D2 Problem of C.T.C. Wall and the Relation Gap problem. I will explain a pro-p version of the conjecture, as well as its higher dimensional abstract analogues and why we can verify the conjecture in these cases.
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The quantum Yang Baxter equation arose in statistical mechanics around 1970 as the consistency condition for an interaction of two particles on a line. In the 1980s, Drinfled and Jimbo introduced quantum groups (deformations of universal enveloping algebras of Lie algebras), and showed that they allow a universal R matrix - an element constructed from the algebra, which systematically produces a solution of the quantum Yang Baxter equation in every representation of this algebra. In turn, this imposes a structure of a braided tensor category on representations of the quantum group (i.e. gives an action of the braid group of type A) and leads to the Reshetikhin-Turaev construction of invariants of knots, braids, and ribbons.
Considering the same problem with a boundary (on a half line instead of a line) leads to the consistency condition called the (quantum) reflection equation, introduced by Cherednik and Sklyanin in the 1980s. I will explain how, in the joint work with S. Kolb, we use quantum symmetric pairs (Noumi, Sugitani, and Dijkhuizen; Letzter 1990s) to construct a universal K-matrix - an element which systematically produces solutions of the reflection equation. This gives an action of the braid group of type B, endowing the corresponding category of representations with a structure of a braided tensor category with a cylinder twist (as defined by T. tom Dieck, R. Haring-Oldenburg 1990s).
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The mathematical structure of quantum moduli spaces in string theory contains a wealth of information about the physical behaviour of the effective field theories. However, research in this area has also lead to very interesting new mathematical structures. In this seminar I will describe new geometrical structures appearing in the context of “heterotic strings” associated to gauge bundles on manifolds with certain special structures. We will see how to recast these geometric systems in terms of the existence of a nilpotent operator and describe the tangent space to the moduli space. I will talk about a number of open problems, in particular, the efforts to understand higher order deformations, the global structure of the full moduli space, and the expectation of new dualities similar to mirror symmetry.
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The close relationship between index theory and representation theory is a classical theme. In particular, the trace formula has been studied through the lens of index theory by several researchers already. In joint work with Bram Mesland (Leiden) and Hang Wang (Shanghai), we take this connection further and obtain a formulation of the trace formula in K-theoretic terms. The central object here is the K-theory group of the C*-algebra associated to a locally compact group. This work is part of a program which explores the potential role that operator K-theory could play in the theory of automorphic forms.
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One of the central objects of interest in additive combinatorics is the sumset $A + B := \{ a+b : a \in A, \, b \in B \}$ of two sets $A,B \subset \mathbb{Z}$. Our main theorem, which improves results of Green and Morris, and of Mazur, implies that the following holds for every fixed $\lambda > 2$ and every $k \ge (\log n)^4$: if $\omega \to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \le \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$. This is joint work with Marcelo Campos, Mauricio Collares, Rob Morris and Victor Souza.
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Expander graphs are somewhat contradictory geometric objects that have many applications, even outside of pure mathematics. We will see how they can be constructed with the help of geometric group theory, and how one can use some coarse-geometric variants of notions from topology to explore the world of resulting constructions.
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Let X be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their X-by-X matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of X. This algebra "encodes" the large-scale (a.k.a. coarse) structure of X. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work. (Joint with A Tikuisis and J Zhang.)
In the talk, I will introduce basics of coarse geometry, Property A and Roe algebras. Then I will move on to quasi-locality and (hopefully) the main ingredients of our argument: If X has Property A, then any quasi-local operator actually belongs to the Roe algebra.|
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Sparse operators are positive dyadic operators that have very nice boundedness properties. The L^p bounds and weighted L^p bounds with sharp constant are easy to obtain for these operators. In the recent years, it has been proven that singular integrals (cancellative operators) can be pointwise controlled by sparse operators. This has made the sharp weighted theory of singular integrals quite straightforward. The current efforts focus in understanding the use of sparse operators to bound rougher operators, such as oscillatory integrals. Following this direction, our goal in this talk is to describe the control of Bochner-Riesz operators by sparse operators.
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Groups of surfaces isogenous to a higher product of curves can be characterised by a purely group-theoretic condition, which is the existence of a so-called ramification structure. Gul and Uria-Albizuri showed that quotients of the periodic Grigorchuk-Gupta-Sidki groups, GGS-groups for short, admit ramification structures. We extend their result by showing that quotients of generalisations of the GGS-groups also admit ramification structures, with some deviations for the case p=2. This is joint work with Elena Di Domenico and Sukran Gul.
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I will discuss a classical problem in Number Theory concerning the distribution of primes in short intervals and explain how an analogue of this problem involving polynomials can be solved by evaluating certain matrix integrals. I will also explain a generalisation to other arithmetic questions with a similar flavour.
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The idempotent conjecture is that there should be no idempotent in the group ring of a torsion-free group. I will discuss this conjecture, as well as associated conjectures in some geometric context, and will use them as an excuse to discuss hyperbolicity and introduce relative hyperbolicity, a context in which some of these conjectures are still open.
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A cover of a space by open subsets is amenable if these subsets all induce amenable images on the level of the fundamental group. In analogy with the LS-category, one can ask how small of an amenable cover one can find for a given space. By Gromov's vanishing theorem, simplicial volume is an example of an obstruction against the existence of small amenable covers. In this talk, I will put this result into context and I will briefly sketch an alternative proof for the vanishing theorem (joint work with Roman Sauer).
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Intuitively a bundle of algebras is a collection of algebras continuously parametrised by a topological space. In operator algebras there are (at least) two different definitions that make this intuition precise: Continuous C(X)-algebras provide a flexible analytic point of view, while locally trivial C*-algebra bundles allow a classification via homotopy theory. The section algebra of a bundle in the topological sense is a C(X)-algebra, but the converse is not true. In this talk I will compare these two notions using the classical work of Dixmier and Douady on bundles with fibres isomorphic to the compacts as a guideline. I will then explain joint work with Marius Dadarlat, in which we showed that the theorems of Dixmier and Douady can be generalized to bundles with fibers isomorphic to stabilized strongly self-absorbing C*-algebras. An important feature of the theory is the appearance of higher analogues of the Dixmier-Douady class.
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Grassmannians are objects of great combinatorial and geometric beauty, which arise in myriad contexts. Their coordinate rings serve as a classical example of cluster algebras, as introduced by Fomin and Zelevinsky at the start of the millennium.
Jensen, King and Su construct an additive categorification of these Grassmannian cluster algebras via maximal Cohen-Macaulay modules over certain plane curve singularities. Such a Grassmannian cluster category encodes key aspects of the cluster structure on the respective coordinate ring of a Grassmannian. Notably, Plücker coordinates naturally correspond to rank 1 modules. An interesting aspect of this relation is that it affords a formal connection between two famous examples of a priori unrelated ADE classifications, providing a bridge between skew-symmetric cluster algebras of finite type and simple plane curve singularities.
In this talk, we take the above ideas to the limit: Taking colimits of Grassmannian cluster algebras gives rise to Grassmannian cluster algebras of infinite rank. We explore these structures combinatorially, and construct an infinite rank analogue of Jensen, King and Su’s Grassmannian cluster categories via maximal Cohen-Macaulay modules over certain hypersurface singularites – the Grassmannian categories of infinite rank. In particular, we investigate how Plücker coordinates are in natural correspondence with generically free modules of rank 1.
This talk is based on joint work with Grabowski, and with August, Cheung, Faber, and Schroll.|
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Modular forms are holomorphic functions on the upper half-plane which display some symmetry with respect to the action of a subgroup of $SL(2,\mathbb{Z})$. However, it turns out that they encode a great deal of arithmetic information about some field extensions of the rational numbers. This relationship has been fruitfully exploited to prove results in number theory, perhaps the more notorious being the proof of Fermat's Last Theorem by A. Wiles.
In this talk we want to describe the interplay between these two subjects and provide an application of field arithmetic to the existence of certain families of weight one modular forms. This is joint work with François Legrand and Gabor Wiese.
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Start with a two-variable complex polynomial f with an isolated critical point at the origin. We will survey a range of classical structures associated to f, and explain how these can be revisited and enhanced using insights from symplectic topology. No prior knowledge of singularity theory or symplectic topology will be assumed.
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The Fourier transform provides a map between function spaces on a given abelian group G and function spaces on its dual group, which interchanges convolution and pointwise product. The functions on G that correspond to integrable functions on its dual group form a natural Banach algebra, known as the Fourier algebra of G. In the 1960s it was shown that one can extend the definition of the Fourier algebra to non-abelian groups, and the resulting Banach algebra has since been the subject of much study. In many cases there is also a corresponding version of the Fourier transform, but scalar-valued Fourier coefficients must be replaced by operator-valued Fourier coefficients.
In this talk, which will mostly be expository, I will give a sketch of these constructions, focusing on some specific examples arising from groups such as SU(2) or the real ax+b group. I will then discuss the following natural but slightly ill-posed question: what operation on the "dual side" corresponds to pointwise product of functions in G? In particular, I will report on recent work (joint with M. Ghandehari) where we are able to describe the dual convolution explicitly for the real ax+b group. Time permitting, I will mention some applications to the study of derivations and cocycles on certain Fourier algebras.
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Over the past decade, research at the interface of topology and neuroscience has grown remarkably fast. Topology has, for example, been successfully applied to objective classification and digital synthesis of neuron morphologies, to automatic detection of network dynamics, and to the construction of a powerful and parameter-free mathematical framework for relating the activity of a network of neurons or brain regions to its underlying structure, both locally and globally.
In this talk I will present a medley of recent applications of topology to neuroscience, many of which resulted from close collaboration with the Blue Brain Project.|
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The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations. This is joint work with A. Carbery.
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Roughly speaking, a ghost operator is often an infinite matrix such that its matrix entries vanish at the infinity. This notion was introduced by Guoliang Yu in the study of the so-called coarse Baum-Connes conjecture. It is a very central topic in coarse geometry and operator algebras with applications to provide counterexamples to the coarse Baum–Connes conjecture, the existence of non-exact groups and the rigidity problem for Roe-type algebras. In this talk, we will visualize a class of ghost projections in terms of expanderish graphs.
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Suppose M is a closed Riemannian manifold with an orthonormal basis B of $L^2(M)$ consisting of Laplace eigenfunctions. Berry's Random Wave Conjecture tells us that under suitable conditions on M, in the high energy limit (ie, large Laplace eigenvalue) elements of B should roughly behave like random waves of corresponding wave number. A classical result of Shnirelman and others that $M$ is quantum ergodic if the geodesic flow on the cotangent bundle of $M$ is ergodic, can then be viewed as a special case of this conjecture.
We now want to look at a level aspect, namely, instead of taking a fixed manifold and high energy eigenfunctions, we take a sequence of Benjamini-Schramm convergent compact Riemannian manifolds together with Laplace eigenfunctions f whose eigenvalue varies in short intervals. This perspective has been recently studied in the context of graphs by Anantharaman and Le Masson, and for hyperbolic surfaces and manifolds by Abert, Bergeron, Le Masson, and Sahlsten. In my talk I want to discuss joint work with F. Brumley in which we study this question in higher rank, namely sequences of compact quotients of $SL(n,R)/SO(n)$ for $n>2$.
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Founded by Chas and Sullivan's observation that the homology of the free loop space of an oriented manifold has the structure of a Batalin--Vilkovisky algebra, string topology studies the rich algebraic structure present on the homology of the free loop spaces of certain spaces such as manifolds and classifying spaces of compact Lie groups. In this talk, I will provide a gentle and subjective introduction to the subject, and also indicate how it connects with objects such as moduli spaces of Riemann surfaces, automorphism groups of free groups, and finite groups of Lie type.
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Coarse geometry studies the large-scale properties of metric spaces, groups and other mathematical objects. Interesting invariants are constructed using coarse homology theories. In this exposition I will explain an axiomatic approach to coarse homology theories. A motivic statement is a statement of the form: For every coarse homology theory E assertion P(E) holds. For example, one can turn the coarse Baum-Connes conjecture into a motivic statement. I will explain how motivic statements can be captured in terms of a universal coarse homology theory. The talk is based on joint work with Alexander Engel.
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An invariant that has attracted quite some attention in the last decade is the magnitude of a compact metric space. Magnitude gives a way of encoding the size of a metric space, resembling both the Euler characteristic and the capacity. In this colloquium I will give a short introduction to magnitude and present some recent results for compact metric spaces of geometric origin (i.e. domains in Euclidean space or manifolds). One of the results states that the magnitude recovers geometric invariants such as volume and certain integrals of curvatures. Based on joint work with Heiko Gimperlein and Nikoletta Louca.
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A central question in quantum information theory is to determine physical resources required for quantum computational speedup. Such resources are characterized in terms of intrinsic features of quantum states and include various notions such as quantum contextuality, quasiprobability representations, and topological phases. Each of these notions correspond to a different perspective taken on the question of where the computational power is hidden. We take a topological approach based on the recently established connection between classifying spaces from algebraic topology and the study of quantum contextuality from quantum foundations in joint work with Robert Raussendorf. In this talk I will explain this connection and discuss possible ways of extending the role of topology to study other kinds of quantum resources.
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I'm planning to talk about some quite old joint work with Ivan Smith which realises moduli spaces of quadratic differentials on Riemann surfaces as spaces of stability conditions on a certain class of three-dimensional Calabi-Yau triangulated categories. I expect to spend the whole talk explaining what all those words mean, and why such a result might be interesting!
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In this talk we will mainly focus on rational homology balls: their history, interest and prominence in nowadays low dimensional topology. We will start with the basic definitions and we will spend some time trying to understand the importance of these balls and how they relate to seemingly disjoint problems. We will end by discussing some recent results which will hopefully give a picture of the current state of the art. No prior knowledge of the topic will be assumed.
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This is a report on joint work with Eva Höning.
For rings of integers in an extension of number fields there are classical methods for detecting ramification and for identifying ramification as being tame or wild. Noether's theorem characterizes tame ramification in terms of a normal basis and tame ramification can also be detected via the surjectivity of the norm map. We take the latter fact and use the Tate cohomology spectrum to detect wild ramification in the context of commutative ring spectra. I will discuss several examples in the context of topological K-theory and modular forms.
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The existence of quantum field theories in higher dimensions predicts many hidden algebraic structures in geometry and topology. In this talk, I will survey some recent developments where such algebraic structures lead to new insights into 1) the quantization of moduli spaces of Higgs bundles, 2) the categorification of quantum invariants of 3-manifolds, and 3) novel types of TQFTs in four dimensions.
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The conjectural local Langlands correspondence connects representations of p-adic groups to certain representations of Galois groups of local fields called Langlands parameters. In recent joint work with Dat, Helm, and Moss, we have constructed moduli spaces of Langlands parameters over Z[1/p] and studied their geometry. We expect this geometry is reflected in the representation theory of the p-adic group. Our main conjecture “local Langlands in families” describes the GIT quotient of the moduli space of Langlands parameters in terms of the centre of the category of representations of the p-adic group generalising a theorem of Helm-Moss for GL(n). I will give an introduction to this picture and explain how after inverting the "non-banal primes" one can prove this conjecture for the local Langlands correspondence for classical groups of Arthur and others.
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The McKay correspondence is a (natural) correspondence between the (non-trivial) irreducible representations of a finite subgroup G of $SL(2,\mathbb{C})$ and the irreducible components of the exceptional divisor of a minimal resolution of the associated quotient singularity $\mathbb{C}^2//G$. A geometric construction for this correspondence was given by González-Sprinberg and Verdier, who showed that the two sets also correspond bijectively to the set of indecomposable reflexive modules on the quotient singularity. This was generalized to higher dimensional quotient singularities (i.e., quotient of $\mathbb{C}^n$ by a finite subgroup of $SL(n,\mathbb{C})$) by Ito-Reid, where the above sets were substituted by certain smaller subsets. It was further generalized to more general quotient singularities by Bridgeland-King-Reid, Iyama-Wemyss and others, using the language of derived categories. In this talk, I will survey past results and discuss what happens for the isolated Q-Gorenstein singularities case (not necessarily a quotient singularity). If time permits, I will discuss applications to Matrix factorization. This is joint work in progress with J. F. de Bobadilla and A. Romano-Velazquez.
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The group Top(d) of homeomorphisms of d-dimensional Euclidean space is a basic object in geometric topology, with its quotient Top(d)/O(d) by the subgroup of linear isometries completely controlling the difference between smooth and topological manifolds in all dimensions (except 4). I will explain some of the classical methods for studying the topology of this group, and report on some recent advances.
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The idea of mutating an object in some family into another has spread across algebra, geometry and combinatorics in the past decade. Some instances have been studied for a long time but the idea has gained new life following the introduction of cluster algebras at the turn of the century. In this talk, I will talk about a framework under development with my collaborators, that uses a “tropical lens” to find and formalise the similarities between the different notions of mutation and which is leading to new understanding of these phenomena.
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Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker, we use algebraic geometry over F1, the so-called field with one element, to construct such moduli spaces. As an application, we streamline various results of matroid theory and find simplified proofs of classical theorems, such as the fact that a matroid is regular if and only if it is binary and orientable.
We will dedicate the first half of this talk to an introduction of matroids and their generalizations. Then we will outline how to use F1-geometry to construct the moduli space of matroids. In a last part, we will explain why this theory is useful to simplify classical results in matroid theory.
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In classical algebra, the integer primes p help decompose objects as well as problems into their p-primary parts, which may be easier to study. The same is true in homotopy theory, but the situation is more interesting since for each integer prime p, there are infinitely many nested homotopical primes. For each of those homotopical primes, there is an (unramified) Galois group that governs the local story and encodes the symmetries of chromatic homotopy theory. These Galois groups turn out to be particularly nice profinite groups, known as compact p-adic analytic. Such groups and their fascinating duality properties within algebra were studied by Lazard. I will try to explain a newer result, which shows that their homotopical duality properties are even better, giving powerful implications for the chromatic Galois extensions that they govern.
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Clemens' conjecture states that the the number of rational curve in a generic quintic threefold is finite. If it is false we prove that certain periods of rational curves in such a quintic threefold must vanish. Our method is based on a generalization of a proof of Max Noether's theorem using infinitesimal variation of Hodge structures and its reformulation in terms of integrals and Gauss-Manin connection.
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Complex singular differential equations play a major role far beyond pure mathematics: classical equations like Airy, Bessel, and Schrödinger equations are just some famous examples that appear in physics and engineering. As geometers, we prefer to study their geometric generalisations called meromorphic connections on holomorphic vector bundles over Riemann surfaces (a.k.a. "differential equations on steroids"). There are many excellent reasons to do this: one is that their moduli spaces (i.e., spaces of equivalence classes) have an incredibly rich geometry that links with a vast variety of subjects from (to name just a few) integrable systems and Poisson geometry, to quantum algebras and representation theory, to Gromov-Witten theory and quantum field theory. Moduli spaces of connections are exceptionally captivating objects, "a gift to geometry that keeps on giving" as some of us would say.
In broad and as accessible terms as possible, I will present a little bit of this really fascinating story to give you a sense or a glimpse of the subject’s richness. At the end, I will mention a word or two about a new geometric method (called abelianisation) to analyse higher-rank connections (i.e., higher-order differential equations) by placing them in correspondence with much simpler objects: rank-one connections (i.e., first-order differential equations) but over a geometrically more complicated Riemann surface.
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Automorphic representations are some of the richest and most mysterious mathematical objects discovered to-date. They simultaneously generalize (i) infinite-dimensional representations of real Lie groups, (ii) modular forms and (iii) the Hecke characters of class field theory. As such, automorphic representations incorporate representation theory, analysis and arithmetic.
In the late 1960's, Robert Langlands laid out a program to unravel much of the seemingly hidden structure of automorphic representations. To begin to understand the Langlands program, it is useful -- at least at first -- to distinguish two kinds of conjectures: Roughly, Langlands' Functoriality Principle can be seen as intrinsic to automorphic representations -- revealing a myriad of relations between different automorphic representations of different groups. By contrast, the extrinsic Langlands correspondence explains how certain automorphic representations should be related to Galois theory and algebraic geometry. Every automorphic representation has associated numerical invariants called Hecke eigenvalues -- these are complex numbers. One of the most interesting aspects of the Langlands program is that some automorphic representations have Hecke eigenvalues which are algebraic numbers, while for others they are transcendental. At this time, we seem to lack a conceptual understanding for why this dichotomy exists. While the Langlands correspondence suggests that certain automorphic representations should have algebraic Hecke eigenvalues, it remains unclear -- even at the level of conjectures -- wherein lies the watershed line between algebraic and transcendental.
I will spend most of my talk introducing automorphic representations, their Hecke eigenvalues, functoriality and the correspondence. The end goal of my talk is then to explain what can be said about the algebraicity of Hecke eigenvalues by combining (1) Previously known cases of algebraicity and (2) Langlands functoriality. On the one hand, I will explain why the algebraicity of Hecke eigenvalues does propagate from some cases to others via functoriality -- this gives new theorems and conjectures on algebraicity of Hecke eigenvalues. On the other hand, I will explain why most cases -- including Maass forms -- are not reducible to known ones via functoriality.
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One of the original motivations of Murray and von Neumann introducing operator algebras was to study the unitary representation theory of groups. This naturally leads to the question of studying building blocks of representation theory, that is simple operator algebras associated with groups. From a modern point of view, not only groups but also other group-like structures such as groupoids should be investigated. This talk introduces the audience to group and groupoid operator algebras and tells the story of how our point of view on their simplicity changed dramatically over the past 10 years. At the end of the talk, I will present some recent results on simple groupoid C*-algebras that were obtained in joint work with Kennedy, Kim, Li and Ursu.
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We describe recent results in quantum or noncommutative Riemannian geometry based on bimodule connections. Here the coordinate algebra can be any unital algebra A equipped with a differential structure expressed as a bimodule Omega^1 of 1-forms as part of a differential graded algebra with A in degree 0. The simplest case is A the commutative algebra of functions on the vertices of a directed graph with Omega^1 spanned by the arrows. We show in this framework that the intrinsic quantum Riemannian geometry of the A_n graph o-o-…-o of n vertices is necessarily q-deformed with q^{2(n+1)}=1. It's q-> 1 limit is the intrinsic quantum Riemannian geometry of the natural numbers viewed as a half-line graph. We then discuss more generally how solutions of the Yang-Baxter or braid relations arise naturally from noncommutative differential geometry and relate both to quantum jet bundles and to the notion of a quantum geodesic.
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In my thesis, I introduced a Floer theoretic invariant for compact subsets of symplectic manifolds called relative symplectic cohomology. This invariant has already proved to be very useful in symplectic rigidity questions and also opened the way to a fruitful reinterpretation of mirror symmetry. Most of these applications rely on an analogue of Mayer-Vietoris property from topology that holds for relative symplectic cohomology under well-understood geometric assumptions. I will briefly introduce the invariant, discuss the Mayer-Vietoris property and present some computations relevant to mirror symmetry. I will try to make the talk accessible to a more diverse audience by mainly sticking to dimension two, where a symplectic form is nothing but an area form.
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A common theme in modern Number Theory is to find interesting discrete objects coming from very different places, but whose arithmetic properties are intimately connected. In this talk we will (hopefully) see a surprising example of this, connecting the first and third objects in the title (using the second to bridge the gap). Time permitting, we will sketch the proof, motivated by a hidden 1.5th object (Clifford algebras). (Based on joint work with E. Assaf, C. Ingalls, A. Logan, S. Secord and J. Voight)
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A measure preserving action of a group G on a measure space X gives rise to a unitary representation of G on the Hilbert space $L^2(X)$. This action may or may not have the spectral gap property which is a very strong form of ergodicity. For instance, groups with Kazhdan's property (T) always have this property. We will survey the importance of the spectral gap property in various problems arising in graph theory, dynamical systems or operator algebras. In the case where X is a homogeneous space arising from an algebraic group, we will show that the absence of the spectral gap property is often related to amenability.
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Theta correspondence is a major theme in the theory of automorphic forms and in representation theory. In a nutshell, the correspondence sets up a bijection between certain sets of smooth admissible irreps of a pair of reductive groups G,H which sit as each others' centralizers in a larger symplectic group.
In joint work with Bram Mesland (Leiden), we showed that the theta correspondence, in many cases, can be interpreted within the framework of Rieffel's induction theory for representations of C*-algebras. This interpretation reveals some new fundamental features: the theta correspondence is functorial and is continuous with respect to weak containment. In the talk, I will explain our approach and time permitting, will discuss some further applications. Many of the results I will discuss can be found in the preprint arXiv:2207.13484.
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One dimensional quantum field theory, also known as quantum mechanics, has been completely understood since the first half of the 20th century, thanks to groundbreaking work of John von Neumann. Two dimensional quantum field theory (2d QFT), on the other hand, has been axiomatised in a variety of different ways, and is still very much work in progress.
An important conjecture known as the Segal-Stolz-Teichner conjecture, predicts that the space of all supersymmetric 2d QFTs has the homotopy type of elliptic cohomology. This conjecture is currently far out of reach. But its consequences are being explored by physicists, and they are finding ample computational evidence for it. Still, it noteworthy that this conjecture has not even reached the stage of precise mathematical conjecture, because of lack of a suitable precise definition of 2d QFT.
A much simpler variant of the Segal-Stolz-Teichner conjecture (still completely out of reach), predicts that the space of all 2d QFTs –not supersymmetric– is contractible. I will present some speculations around this latter conjecture, and use them to make some physical predictions.
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In three different areas of Mathematics (Commutative Algebra, Algebra of differential operators, and Poisson algebras) there are three long standing open conjectures that turned out to be equivalent: the Jacobian Conjecture, the Dixmier Conjecture, and the Poisson Conjecture. These are questions about whether certain homomorphisms which are "almost automorphisms" are in fact automorphisms.
We show that the Jacobian Conjecture, the Dixmier Conjecture, and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra $A_n$. Using this approach we show that the images of the Jacobian maps, endomorphisms of the Weyl algebra $A_n$ and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples (the conjectures hold if and only if the images coincide with the algebras). A short direct algebraic (without reduction to prime characteristic) proof is given of equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures).
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The Birch and Swinnerton-Dyer conjecture (BSD) plays a pivotal role in the study of elliptic curves by allowing us to solve their equations systematically. In this talk, I will first define elliptic curves and present the conjecture. Then I will explain how BSD beautifully blends key arithmetic information of the curve into a surprising formula and will introduce the Parity conjecture. I’ll discuss a few results on the Parity conjecture in collaboration with V. Dokchitser and H. Green.
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This story starts with rotations in Euclidean 3-space: the finite subgroups of SO(3) are either cyclic or dihedral or one of the symmetry groups of the Platonic solids. In the 19th century, Felix Klein investigated the orbit spaces of those groups and their double covers, the so-called binary polyhedral groups. This investigation is at the origin of singularity theory.
Quite surprisingly, in 1979, John McKay found a direct (though then mysterious) relationship between the resolution of the singularities of the orbit spaces and the representation theory of the finite group one starts from. This "McKay correspondence" is manifested, in particular, by the ubiquitious Coxeter-Dynkin diagrams. The McKay correspondence marks essentially the beginning of "Noncommutative singularity theory", the use of representation theory of not necessarily commutative algebras to understand the geometry of singularities, a subject area that has exploded during the last two decades in particular because of its role in the mathematical formulation of string theory in Physics.
In this talk I will survey the beautiful classical mathematics at the origin of this story and then give a sampling of recent results (joint with Ragnar-Olaf Buchweitz, Colin Ingalls, Simon May, and Marco Talarico) and work still to be done.
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One of the first theorems that we prove in the complex representation theory of a finite group is that the number of irreducible representations (up to isomorphism) equals the number of conjugacy classes in the group. For example, for the group of permutations of the set {1,2,...,n}, the conjugacy classes are parametrised by partitions of n (the cycle decomposition) and so are the complex irreducible representations (via Young's construction from the 1890s). But this is not a natural bijection, just like there is not a natural isomorphism between a finite vector space and its dual in general. However, for certain classes of groups, that come with extra structure (like the ones appearing in Lie theory), one expects natural relations between the irreducible representations of the group, on one hand, and conjugacy classes in a *dual* group, on the other. This happens for example, when the group in question is a finite reflection crystallographic group, or a connected algebraic group over a finite or local field. In these correspondences, a particularly interesting role is played by the unipotent conjugacy classes in the dual group. I will give a survey of some of these connections and then emphasise the case of (infinite-dimensional) representations of reductive algebraic groups (like the general linear group of n by n matrices) with coefficients in a local field, where I'll explain what the unipotent classes tell us about the growth of characters and the parametrisation of such representations. The new results in the talk are joint with Lucas Mason-Brown and Emile Okada.
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Modular symbols are a fundamental tool for the computation of the homology of certain linear groups. It has been observed that, surprisingly, they also control the relations among certain trigonometric and elliptic functions. After introducing modular symbols and their elementary properties I will explain why this is the case and give some arithmetic applications.
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In ring theory, Morita equivalence preserves many properties of the objects, and generalizes the isomorphism equivalence between commutative rings. A strong Morita equivalence for selfadjoint operator algebras was introduced by Rieffel in the 60s, and works as a correspondence between their representations. In the past 30 years there has been an interest to develop a similar theory for nonselfadjoint operator algebras and operator spaces with much success. Taking motivation from recent work of Connes and van Suijlekom, we will present a Morita theory for operator systems. We will give equivalent characterizations of Morita equivalence via Morita contexts, bihomomoprhisms and stable isomorphism, while we will highlight properties that are preserved in this context. Time permitted we will provide applications to rigid systems, function systems and non-commutative graphs. This is joint work with George Eleftherakis and Ivan Todorov.
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In this talk, I will explain how to test efficiently and effectively whether two odd modular Galois representations of the absolute Galois group of the rational numbers are isomorphic. In particular, I will present new optimal bounds on the number of traces to be checked (joint work with Peter Bruin, University of Leiden). I will also briefly discuss graphs of isomorphisms associated to such objects, related results on Hecke algebras, and a database of modular representations.
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In this talk, we will present some results concerning the spectrum of Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichmüller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl’s law is satisfied in terms of critical exponents of the representation which are completely determined by the Manhattan curve associated to the Teichmüller deformation. This is joint work with Frédéric Naud.
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Ambidexterity refers to a situation where we have two functors $F\colon\mathcal{C}\to\mathcal{D}$ and $G\colon\mathcal{D}\to\mathcal{C}$ such that $F$ is both left adjoint and right adjoint to $G$. There are elementary examples involving complex representation theory of finite groups and groupoids, and deep examples involving chromatic cohomology of finite $\infty$-groupoids (or equivalently, $\pi$-finite spaces), as well as various other examples of intermediate depth. I will survey some of these ideas, with emphasis on applications in chromatic stable homotopy theory.
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Fix a finite set R of positive integers bigger than one with no common factors. The Frobenius number for R is the largest number that cannot be written as a linear combination of the integers in R with non-negative integral coefficients.
In general, Frobenius numbers fluctuate. To study such things, we search for structures. Here, the given set of positive integers R can be a point in the lattices studied in the dynamics and number theory crossover. We study the behaviour of these rational points on expanding closed horospheres in the space of lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira (2016). Their proof uses techniques from homogeneous dynamics and relies particularly on measure-classification theorems, due to Ratner. We pursue an alternative strategy based on Fourier analysis, Weil's bound for Kloosterman sums, recently proved bounds (by M. Erdélyi and Á. Tóth) for matrix Kloosterman sums, Roger's formula, and the spectral theory of automorphic functions.This is a joint work with D. El-Baz, B. Huang, J. Marklof and A. Strömbergsson.
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In 1980 and 1981, Kaufman constructed measures with polynomial Fourier decay on the set of badly-approximable numbers and the set of well-approximable numbers. The strategy for the badly-approximable numbers uses the continued fraction expansion together with a change-of variables, and the strategy for the well-approximable numbers uses the cancellation of an exponential sum. We will discuss the application of both of these strategies to the set of numbers approximable to exact order introduced by Bugeaud. This talk is based on joint work with Reuben Wheeler.
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In mathematics, we like classifying objects. A moduli space is a space where each point represents a(n isomorphism class of a) space satisfying certain criteria, giving a geometric answer to a classification problem. Often the geometry of such spaces are interesting in our own right and their corresponding enumerative information has rich structure. We will study the case of genus-zero n-pointed curves and a generalisation where they are further equipped with an r-spin structure. Enumerative invariants built from their characteristic classes have rich structure due to generalisations of predictions of Witten confirmed by Kontsevich. We will explain approaches to understanding these invariants on a very concrete level through combinatorial structures like recursion and tropical geometry.
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An infinite sequence $a = (a_n)_{n\geq 0}$ is $q$-automatic if an is a finite-state function of the base-$q$ expansion of $n$. This means that there exists a deterministic finite automaton that takes the base-q expansion of $n$ as input and produces the symbol an as output for each $n \in \mathbb{N}$.
Automatic sequences appear in diverse fields of mathematics, such as algebra, logic, number theory, and topological dynamics. They have the advantage of lend- ing themselves to computation, so that in each area there arise specific problems concerning automatic sequences, and much of the time, constructive solutions.
I will give a background of their characterisations in algebra and dynamics, via Furstenberg’s, Cobham’s and Christol’s theorems. I will then talk about joint work with Eric Rowland and Manon Stipulanti, concerning automatic sequences in number theory, and also about joint work with Johannes Kellendonk, concerning automatic sequences in topological dynamics, ending with a topological invariant which seems to defy computation.
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Given a compact Kähler manifold satisfying an integrality condition, the Berezin-Toeplitz geometric quantization construction produces matrix algebras; these fit together into a fundamental example of strict deformation quantization. The integrality condition can be circumvented by passing to the universal covering space, if the lift of the symplectic form is exact; in this case, the symplectic form determines a 2-cocycle of the fundamental group. The key to analyzing this construction is to use Hilbert $C^*$-modules, which generalize Hilbert spaces. The resulting algebras are more interesting than matrix algebras and are partially determined by index theorems. The simplest example is the noncommutative torus, and this gives higher-genus noncommutative Riemann surfaces as well.
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Diagram algebras, such as the Brauer algebras and Temperley-Lieb algebras, have been studied for many years. They appear in wide-ranging places such as statistical mechanics, knot theory and representation theory. However, the study of the homology of these algebras is a very young field indeed, having emerged over the course of last decade. In this talk I will give an introduction to these diagram algebras, their homology and their connection to group homology and homological stability. Time permitting, I will discuss some recent generalizations of these algebras.
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The Hilbert Scheme of points of n points in the plane is a smooth algebraic variety with a rich topology connected to partitions and representation theory. If G acts on a C^2, it also acts on the Hilbert scheme of points. The question of when certain G fixed point sets are nonempty winds up having a connection to zig-zag permutations, which are counted by the Taylor series coefficients of Tan(x) and Sec(x).
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We will start with a short discussion on semi-classical analysis to introduce the concept of quantum limits. We will present an overview of sub-Riemannian geometry and the recent developments of spectral geometry in this context, especially quantum limits on nilpotent Lie groups.
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I will begin with an introduction to algebraic K-theory, ring spectra and the chromatic redshift conjecture. After this, I will talk about our new proof of the redshift conjecture for Lubin-Tate spectra and our algebraic K-theory computations. This work is partially joint with Christian Ausoni and Tasos Moulinos.
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Some spaces seem to have different dimensions at different scales. A long thin strip might appear one-dimensional at a distance, then two-dimensional when zoomed in on, but when zoomed in on even closer it is seen to be made of a finite array of points, so at that scale it seems zero-dimensional. I will present a way of quantifying this phenomenon using a couple of measures of the size of metric spaces, namely magnitude and spread. I will show lots of examples for finite metric spaces.
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I explain some cancellation and non-cancellation phenomena in algebraic geometry and relate them to the structure of the Grothendieck ring of varieties and to the groups of birational self-maps of algebraic varieties, in particular the Cremona groups.
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In ancient Greece, geometry was about points, lines, circles, and communicated through pictures. The 17th Century marked a transformative shift, connecting geometry with algebra, and lead to working with equations over visual representations. Algebraic geometry emerged as a magical blend of geometric intuition and algebraic methods. Commutative algebra, mainly the study of polynomial rings and their ideals, dominated the field for an extensive period. Then with the emergence of noncommutative algebras, such as matrix algebras, our unstoppable geometric intuition hit an immovable wall. The solution? A return to pictures as representations. In this expository talk, I will introduce a visual perspective on algebras, exploring path algebras and their captivating connections to different fields.
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Aperiodic sequences and sequence spaces form prototypical mathematical models of quasicrystals. The most quintessential examples include subshifts of Sturmian words and substitutions, which are ubiquitous objects in ergodic theory and aperiodic order. Two of the most striking features these shift spaces have, are that they have zero topological entropy and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions, and in stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy and exhibit uncountably many ergodic measures. Moreover, they have been shown to provide mathematical models for physical quasicrystals with defects.
We will begin by talking about subshifts generated by Sturmian words and ways to measure their complexity beyond topological entropy, and show how this measure of complexity can be used to build a classification via Jarník sets. We will then build a bridge between these subshifts and subshifts of random substitutions. We will conclude with some recent dynamical results on subshifts of random substitutions and ways to visualise these subshifts. Namely, we will present a method to build a new class of Rauzy fractals.
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We use 3d defect TQFTs to give a gauge theoretical formulation of (untwisted) Dijkgraaf-Witten TQFT with defects. This leads to a simple description in terms of embedding quivers, groupoids and their representations. Defect Dijkgraaf-Witten TQFTs is then formulated in terms of spans of groupoids and representations of spans. This is work in progress with João Faría-Martins, University of Leeds.
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Over the past decade or so, homotopy type theory (HoTT) has emerged as a novel foundation for mathematics. Rather than taking sets as the basic entities of mathematics, HoTT provides us with a synthetic theory of structures, expressing naturally notions of structural equivalence. These structures are infinity-groupoids, or what Peter Scholze has called ‘anima’. Evidence that the underlying dependent type theory is well-suited to present mainstream mathematics comes from the success of Kevin Buzzard’s program to use Lean as an automated proof assistant to verify contemporary results.
In HoTT itself it is possible to develop what is called ‘synthetic homotopy theory’. But mathematicians also treat further varieties of structure, such as cohesion, smooth structure, equivariance and linear structure. It turns out that these may all be treated synthetically by the addition of ‘modalities’ to HoTT. With the close relationship between HoTT and computation, it appears that Linear HoTT has things to say about quantum computation.
In this talk I shall be giving a gentle introduction to these ideas.
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