Let $G$ be an infinite discrete group; e.g. hyperbolic $3$-manifold group. Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of $G$ alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps.
The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas.
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Dirichlet related the residue at $s=1$ of the Dedekind zeta function of a number field $F$ (a slight generalisation of the famous Riemann zeta function) to two important arithmetical notions: the size of the ideal class group and the `volume' of the unit group in the number ring $O_F$ of $F$. Generalising this surprising connection, the special values of the Dedekind zeta function of a number field $F$ at integer argument n should, according to Zagier's Polylogarithm Conjecture, be expressed via a determinant of $F$-values of the $n$-th polylogarithm function. Goncharov laid out a vast program incorporating this conjecture using properties of multiple polylogarithms and the structure of a motivic Lie coalgebra.
In this impressionist talk I intend to give a rough idea of the developments from the early days on, avoiding most of the technical bits, and also hint at a number of recent results for higher weight, some in joint work with, or developed by, S.Charlton, D.Radchenko as well as D.Rudenko and his collaborators.
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Matroids are combinatorial structures that track ``independence'' relations on a set. A key example is linear independence of some linear functions on a vector space. Not all matroids come from a vector space, but those that don't behave in surprising algebraic ways as if they do. Breakthroughs of the last decade have opened a kit of tools from, and inspired by, algebraic geometry to prove inequalities for matroids, among them the ``matroid Hodge theory'' of June Huh and others. I'll start by motivating matroids, and aim to end with enough about my work in progress with Andy Berget to show how its central tool is different to matroid Hodge theory.
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Complex dynamics began in the early 1900s with the study of iterating polynomial functions with complex coefficients. This simple idea gives rise to beautiful fractal pictures such as the Mandelbrot set, as well as interesting mathematical questions of many different flavours (algebraic, analytic, topological, arithmetic, etc.). The field gained momentum in the 1980s due to work of Thurston, Douady-Hubbard, Sullivan, and others, connecting these dynamical questions to surface topology and the theory of 3-manifolds. The last decade has seen many breakthroughs achieved via new tools from number theory, measure theory and algebraic geometry. I will discuss some of these recent developments, highlighting the interplay between topology on one hand and algebraic geometry/number theory on the other hand.
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I give an overview of a project with Michael Wemyss to classify simple 3-fold flops. This amounts to understanding when a smooth rational curve (i.e. the Riemann sphere) inside a complex 3-dimensional manifold can be contracted. (One might think of this as a 3d analogue of shrinking the central axis of a Moebius strip to a point, and indeed one could do an unnecessarily elaborate analysis of that situation by the same methods.) In fact, this large family of surgery operations is central to 3d complex geometry, but has nevertheless resisted classification, or even the construction of a set of representative examples.
Being a manifold, one can describe the situation by glueing together patches - and it is enough to glue together two copies of the affine space C^3 by a simple formula .. but with lots of free parameters, most of which do not contract. However finding good (i.e. contractible) glue functions (or even classifying them) seems to be a bit needle-in-a-haystack. Instead, we translate the problem to one of classifying noncommutative germs f(x,y) [or equivalently certain complete local algebras up to isomorphism], where the necessary criteria seem more amenable. That context feels much like the classical singularity theory of function germs in the style of Arnold (types ADE and all that), and we can solve enough of that problem to construct all flops and to provide a classification.
From one point of view, I’d like to give some idea of what Theorems 5.1 and 5.4 of the following expository tea-time article mean: https://arxiv.org/abs/2410.21500
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This is joint work with my PhD student Tom Lawrence.
For a prime p, the p-decomposition matrix D of a finite group G records the way each irreducible ordinary representation of G breaks up into irreducible p-Brauer characters under reduction modulo p. Multiplying D by its transpose yields the Cartan matrix, whose determinant is well-known to be a power of p. A representation of a Sylow p-subgroup S of G is fusion-stable if it is left invariant by the conjugation action of G. After first fixing a basis B of fusion-stable representations of S one can consider an analogue of D for fusion-stable representations which records how each irreducible ordinary representation of G breaks up in B under restriction to S. It turns out this matrix has many properties analogous to those of the classical decomposition matrix, and using them one can show that the modulus square of the determinant of the fusion-stable character table (columns indexed by G-classes of p-elements, rows by elements of B) is always a particular power of p independent of the choice of B. I conjectured that the same result holds for any saturated fusion system on S and I'll provide some evidence for this by explicitly computing with some infinite families of exotic examples. If time permits I will also explain how this project fits within the larger framework of "exotic representation theory" whose aim to extend results about ordinary representations to the settings of fusion systems, spetses and other related structures.
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In the first half of the talk, I will explain some recent breakthroughs in the study of the area preserving homeomorphism groups of surfaces using Floer theory. After that, I will explain what happens when we try to generalize it to higher dimensions and the relation to Khovanov homology as well as the Hilbert schemes of points. No prior knowledge on Floer theory or symplectic geometry is assumed.
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