Seminars this semester

Series:

Feb 1

Tue

Samuel W (Sheffield)

Topology Seminar

16:00

I-adic towers and Koszul complexes in algebra and topology

Hicks Seminar Room J11

Feb 8

Tue

Samuel W (Sheffield)

Topology Seminar

14:00

I-adic towers and Koszul complexes in algebra and topology

Hicks Seminar Room J11

Feb 15

Tue

Samuel W (Sheffield)

Topology Seminar

14:00

I-adic towers and Koszul complexes in algebra and topology

Hicks Seminar Room J11

Feb 22

Tue

Samuel W (Sheffield)

Topology Seminar

14:00

I-adic towers and Koszul complexes in algebra and topology

Hicks Seminar Room J11

Mar 1

Tue

Neil Strickland (Sheffield)

Topology Seminar

14:00

Morava K-theory I

Hicks Seminar Room J11

Abstract:
I will give a series of three or four lectures introducing Morava K-theory and Morava E-theory.

Mar 15

Tue

Neil Strickland (Sheffield)

Topology Seminar

14:00

Morava K-theory III

Hicks Seminar Room J11

Abstract:
I will discuss the Morava K-theory of various spaces, such as classifying spaces of finite groups.

Apr 18

Mon

Andrew Stacey (Sheffield)

Topology Seminar

14:00

The Differential Topology of Loop Spaces I

Hicks Seminar Room J11

Abstract:
The aim of these seminars is to provide a gentle but detailed introduction to the study of loop spaces as manifolds. This is a topic which has a long history, dating back at least to the days of Morse, and which has recently received renewed interest due to its strong links with string theory. We shall end this mini-series with an overview of my work on the Dirac operator on loop spaces. This finale will dictate the itinery of the tour: 1. What is an infinite dimensional manifold and how do we know that the loop space is one? 2. What does it look like, what can we do with it, and what do we want to do with it? 3. What's the big deal about Dirac operators in infinite dimensions? It is intended that anyone with basic differential topology should be able to follow these seminars.

Apr 26

Tue

Andrew Stacey (Sheffield)

Topology Seminar

15:00

The Differential Topology of Loop Spaces II

Hicks Seminar Room J11

May 3

Tue

Andrew Stacey (Sheffield)

Topology Seminar

15:00

The Differential Topology of Loop Spaces III

Hicks Seminar Room J11

May 10

Tue

Sarah Whitehouse (Sheffield)

Topology Seminar

15:10

Stable and unstable K-theory operations

Hicks Seminar Room J11

May 17

Tue

Mike Mandell (Cambridge)

Topology Seminar

15:10

A Localization Sequence for the Algebraic K-Theory of Topological K-Theory

Hicks Seminar Room J11

Abstract:
In many ways the algebraic K-theory of ring spectra behaves like the algebraic K-theory of traditional rings. One limitation is the lack of a general formulation of a devissage theorem. Recent work (joint with Andrew Blumberg) establishes one very special case of the devissage theorem. This case is sufficient to construct the localization sequence conjectured by Rognes relating the algebraic K-theory of (complex) K-theory, connective K- theory, and the integers.

May 24

Tue

Ieke Moerdijk (Sheffield)

Topology Seminar

15:00

What do classifying spaces classify?

Hicks Seminar Room J11

May 31

Tue

Neil Strickland (Sheffield)

Topology Seminar

15:10

The Rezk logarithm I

Hicks Seminar Room J11

Abstract:
The Rezk logarithm is a natural map $(E^0X)^\times\rightarrow E^0X$ defined for all spaces $X$ and suitable generalised cohomology theories $E$. In many cases it is close to being an isomorphism. There is a simple definition using a functor constructed by Bousfield and Kuhn, but the thing that makes it usable is a theorem of Rezk relating it to the theory of power operations, and in particular the Hecke operators studied by Ando. This seminar will be the first of a series covering some of this material.

Jun 7

Tue

Neil Strickland (Sheffield)

Topology Seminar

15:10

The Rezk logarithm II

Hicks Seminar Room J11

Abstract:
I will talk about generalized Moore spectra, K(n)-localisation, and the Bousfield-Kuhn functor, all of which are ingredients in the definition of the Rezk logarithm.

Jul 5

Tue

Neil Strickland (Sheffield)

Topology Seminar

15:10

The Rezk Logarithm II'

Hicks Seminar Room J11

Abstract:
I will talk about generalized Moore spectra, K(n)-localisation, and the Bousfield-Kuhn functor, all of which are ingredients in the definition of the Rezk logarithm. This will essentially be a repeat of the seminar I gave a few weeks ago when many people were away.

Oct 4

Tue

Johann Sigurdsson (Sheffield)

Topology Seminar

14:00

Duality in parametrized homotopy theory

Hicks Seminar Room J11

Abstract:
I will describe formal structure enjoyed by the parametrized stable homotopy categories and how one can encode it into a single bicategory. I will then discuss duality theory from that perspective and show how it gives simple conceptual proofs of generalizations of various known duality phenomena such as Atiyah duality and the Wirthmuller and Adams equivalences. The talk should be accessible to everyone.

Oct 11

Tue

Johann Sigurdsson (Sheffield)

Topology Seminar

14:00

Duality in parametrized homotopy theory

Hicks Seminar Room J11

Oct 18

Tue

Simon Willerton (Sheffield)

Topology Seminar

14:00

The derived category of sheaves on a complex manifold from a representation theory perspective

Hicks Seminar Room J11

Abstract:
I will try to explain how the derived category of sheaves on a complex manifold (which I will remind you of) looks a lot like the representation category of a finite group. This will be motivated by ideas from topological field theory.

Oct 25

Tue

David Gepner (Sheffield)

Topology Seminar

14:00

Equivariant elliptic cohomology

Hicks Seminar Room J11

Nov 15

Tue

Andrew Stacey (Sheffield)

Topology Seminar

14:00

Delooping Moravian Maps

Hicks Seminar Room J11

Abstract:
One of the pieces of baggage that comes with a graded cohomology theory is the family of operations. These are self-maps of the cohomology groups obeying certain obvious naturality conditions. There are two main types of operation: stable and unstable. An unstable operation acts only on the cohomology groups of a particular degree whilst a stable operation acts on the cohomology groups of any degree compatibly with the suspension isomorphism. It is clear, therefore, that a stable operation defines a family of unstable ones. However, even if one knows that an unstable operation came from a stable one it may not be easy to reconstruct that stable operation. What is remarkable about the Morava K--theories is that there is a straightforward way to do this. The "delooping" of the title refers to the fact that operations are closely linked to maps between certain spaces and spectra associated to the cohomology theory. In this language, the claim is that there is a simple way to convert an arbitrary map between the representing spaces of the Morava K-theories into an infinite loop map. The mathematics involved is astonishingly simple and I shall endeavour to keep the exposition in a similar vein. Thus the prerequisites are minimal: a familiarity with cohomology theories and their links with spectra. This work is joint with Sarah Whitehouse and is funded as part of the EPSRC project on operations in Morava K--theories.

Nov 22

Tue

Ruben Sanchez (Sheffield)

Topology Seminar

14:00

Classifying spaces for proper actions and the Baum-Connes Conjecture

Hicks Seminar Room J11

Abstract:
I will explain how to generalize the ordinary classifying space of a group G to actions with finite stabilizers. The corresponding classifying space appears in the Baum-Connes Conjecture, which identifies two objects associated to G, one analytical and one topological. The analytical one is the K-theory of the reduced $C^*$-algebra of G, and the topological one is the equivariant K-homology of this classifying space. I will describe how to use Bredon homology and a spectral sequence to obtain the topological side of Baum-Connes. Then I would like to explain how to do this for the groups $SL(3,\mathbb{Z})$ and for some Coxeter groups. The talk may suit two sessions, so if people are not too unhappy, I may also talk the following week.

Nov 29

Tue

Ruben Sanchez (Sheffield)

Topology Seminar

14:00

Equivariant K-homology for $SL(3,\mathbb{Z})$ and Coxeter groups

Hicks Seminar Room J11

Abstract:
I will show how to compute the topological side of the Baum-Connes conjecture for $SL(3,\mathbb{Z})$ and some Coxeter groups. I will put some illustrative pictures.

Dec 13

Tue

Halvard Fausk (Oslo)

Topology Seminar

14:00

t-model structures

Hicks Seminar Room J11

Abstract:
For every stable model category $M$ with a certain extra structure, we produce an associated model structure on the pro-category $Pro(M)$ and a spectral sequence, analogous to the Atiyah-Hirzebruch spectral sequence, with reasonably good convergence properties for computing in the homotopy category of $Pro(M)$. Our motivating example is the category of pro-spectra. The extra structure referred to above is a t-model structure. This is a rigidification of the usual notion of a t-structure on a triangulated category. A t-model structure is a proper simplicial stable model category $M$ with a t-structure on its homotopy category together with an additional factorization axiom.

Oct 5

Thu

Holger Brenner (Sheffield)

Topology Seminar

15:10

Continuous solutions to algebraic forcing equations

Hicks Seminar Room J11

Abstract:
Let $f_1$, ..., $f_n$ and $f$ be polynomials in $C[X_1,...,X_m]$. When is it possible to write $f = q_1f_1 + ... + q_nf_n$ with continuous functions $q_i: C^m \to C$ ($C$=complex numbers). Does there exists an algebraic characterization of this property? The set of polynomials $f$ which can be written in this way form an ideal which we call the continuous closure of $(f_1,...,f_n)$. We give exclusion and inclusion criteria for this closure operation and algebraic apporoximations, in particular in terms of the axes closure (to be introduced). In the case of a monomial ideal we show that the continuous closure and the axes closure have the same combinatorial description and coincide.

Oct 10

Tue

Burt Totaro (Cambridge)

Topology Seminar

14:00

The geometry of Hilbert's fourteenth problem

Hicks Seminar Room J11

Abstract:
All kinds of classification problems in geometry (going back to Euclid) lead to the problem of finding the ring of polynomial invariant functions for a group acting on a vector space. Hilbert asked whether rings of invariants are always finitely generated. The answer is yes in many cases but no in general, by Nagata. Although the problem is formulated algebraically, Nagata's counterexamples make brilliant use of the geometry of algebraic curves. I will present the latest advances on the problem.

Oct 31

Tue

Victor Snaith (Sheffield)

Topology Seminar

14:00

Upper Triangle Technology and the Arf Invariant

Hicks Seminar Room J11

Nov 7

Tue

Alastair Craw (Glasgow)

Topology Seminar

14:00

Hicks Seminar Room J11

Nov 14

Tue

Neil Strickland (Sheffield)

Topology Seminar

14:00

Structured ring spectra and the nilpotence theorem

Hicks Seminar Room J11

Abstract:
One form of the nilpotence theorem says that if $R$ is a ring spectrum and $a\in\pi_dR$ maps to zero in $MU_dR$ then $a^n=0$ for large $n$. This is a very powerful result, which forms the basis for a huge body of work in stable homotopy theory. Strangely, however, little further work has been done with the circle of ideas used in the proof of the nilpotence theorem. In this talk we will revisit these ideas using some newer technology of structured ring spectra.

Dec 5

Tue

Andrew Ranicki (Edinburgh)

Topology Seminar

14:00

The geometric Hopf invariant

Hicks Seminar Room J11

Abstract:
The talk will be a report on an ongoing joint project with Michael Crabb (Aberdeen). The geometric Hopf invariant of a stable map $F:\Sigma^{\infty}X \to \Sigma^{\infty}Y$ is a stable $Z_2$-equivariant map $h(F):X \to (S^{\infty})^+\wedge(Y \wedge Y)$ to the quadratic construction on $Y$. The stable $Z_2$-equivariant homotopy class of $h(F)$ is the primary obstruction to desuspending $F$. The geometric Hopf invariant of the stable Umkehr map $F:\Sigma^{\infty}M^+ \to \Sigma^{\infty}T(\nu_f)$ of an immersion $f:N^n\to M^m$ of manifolds factors through the $Z_2$-equivariant double point set of $f$. The $\pi_1$-equivariant version of the geometric Hopf invariant has an application to Wall's non-simply-connected surgery theory.

Dec 12

Tue

Simon Willerton (Sheffield)

Topology Seminar

14:00

Hopf Monads

Hicks Seminar Room J11

Abstract:
Representations of finite groups have various nice properties, you can tensor two representations together to get another representation and you can take the dual of a representation to get a new representation. This makes the category of representations into a `monoidal category with duals' which lifts these structures from the category of vector spaces. More generally this is true of the representations of any Hopf algebra. A monad is a categorical gadget which can be viewed as generalization of an algebra (in a sense I will explain), and which has a category of representations. Motivated by some specific examples you can ask when the category of representations is a monoidal category with duals (ie when the monad is a *Hopf* monad). I will endeavour to explain my pictorial approach to the answer given by Bruguiere and Virilizier.

Feb 20

Tue

John Greenlees (Sheffield)

Topology Seminar

14:00

Rational cohomology theories on free $G$-spaces

Hicks Seminar Room J11

Abstract:
I aim to describe a classification for the theories of the title. More precisely, they are classified by free rational $G$-spectra, and I will describe an algebraic model when $G$ is a connected compact Lie group (the category of torsion modules over the polynomial ring $H^*(BG;Q)$). The two ingredients are an Adams spectral sequence and derived Morita theory. (Joint work with Brooke Shipley).

Feb 27

Tue

John Greenlees (Sheffield)

Topology Seminar

14:00

Rational cohomology theories on free $G$-spaces pt II

Hicks Seminar Room J11

Mar 5

Mon

Jos (Universidad Nacional Aut)

Topology Seminar

14:00

Characteristic Classes and Transversality

Hicks Seminar Room J11

Abstract:
Let $\xi$ be a smooth vector bundle over a differentiable manifold $M$. Let $h : \epsilon^{n-i+1}\to \xi$ be a generic bundle morphism from the trivial bundle of rank $n-i+1$ to $\xi$. We give a geometric construction of the Stiefel-Whitney classes when $\xi$ is a real vector bundle, and of the Chern classes when $\xi$ is a complex vector bundle. Using $h$ we define a differentiable closed manifold $Z(h)$ and a map $\phi : Z(h)\to M$ whose image is the singular set of $h$. The $i$-th characteristic class of $\xi$ is the Poincaré dual of the image, under the homomorphism induced in homology by $\phi$, of the fundamental class of the manifold $Z(h)$. We extend this definition for vector bundles over a paracompact space, using that the universal bundle is filtered by smooth vector bundles.

Mar 6

Tue

Martin Crossley (Swansea)

Topology Seminar

14:00

Word Hopf Algebras

Hicks Seminar Room J11

Abstract:
Hopf algebras of words appear in many contexts, particularly in topology and in combinatorics. I'll discuss a few of these situations a number of results both old, new, false and true about them.

Mar 13

Tue

Richard Hepworth (Sheffield)

Topology Seminar

14:00

Chen-Ruan Cohomology

Hicks Seminar Room J11

Abstract:
Chen-Ruan cohomology seems to be the correct notion of cohomology for orbifolds. Sadly, the definition is rather complicated since it involves differential operators on Riemann surfaces. I will motivate and define orbifolds and Chen-Ruan cohomology before explaining how all of the complications can be reduced to a single property of the so-called age grading.

Mar 20

Tue

Julia Singer (Bonn)

Topology Seminar

14:00

Equivariant Lambda Rings

Hicks Seminar Room J11

Abstract:
The existence of commutative multiplications on Moore spectra for certain types of rings leads to algebraic conditions providing additional structure on the rings. I'll explain why this can be thought of as an equivariant generalisation of a lambda ring structure.

May 8

Tue

Ruben Sanchez (Sheffield)

Topology Seminar

14:00

Computing Borel's regulator

Hicks Seminar Room J11

Abstract:
The Borel's regulator map is a useful tool to study the higher algebraic K-theory of the ring of integers of an algebraic number field. In 2000, Hamida proved a formula for the Borel's regulator as an integral of non-commutative differential forms. We will present a formula to approximate this integral which can lead to explicit computations. Finally, we will discuss a p-adic version of this.

May 15

Tue

Richard Hepworth (Sheffield)

Topology Seminar

14:00

What is a KO object?

Hicks Seminar Room J11

Abstract:
The real question is "What is an elliptic object?". Stolz and Teichner have been trying to answer this, and along the way they have developed a new perspective on K-theory. In this expository talk I'll try to explain a little bit of this, hopefully ending with a sketch of Stolz-Teichner's theorem describing the KO-theory spectrum in terms of euclidean field theories.

May 21

Mon

Tore Kro (NTNU)

Topology Seminar

14:00

Geometry of elliptic cohomology

Hicks Seminar Room J11

Abstract:
We review what elliptic cohomology is. Furthermore, we will mention the various attempts to define it geometrically. In the program initiated by Baas, the idea is to consider 2-vector bundles. We will look at their definition, and the related notion of charted 2-bundles, and give examples.

May 22

Tue

Tore Kro (NTNU)

Topology Seminar

14:00

What does the nerve of a 2-category classify?

Hicks Seminar Room J11

Abstract:
We outline the proof showing that the nerve of a topological 2-category classifies charted 2-bundles structured by this 2-category. As a corollary, we will see that the K-theory associated to Baez and Crans 2-vector bundles splits as two copies of ordinary K-theory.

May 29

Tue

Ieke Moerdijk (Sheffield)

Topology Seminar

14:00

To what extent is Lie theory for groupoids like that for groups?

Hicks Seminar Room J11

Abstract:
Lie groupoids play an increasingly important role in foliation theory, symplectic and Poisson geometry, and non-commutative geometry. In this lecture, we explain how some basic properties of Lie groups extend to groupoids, and how some other properties don't. The talk will only presuppose some basic familiarity with Lie groups, and in particular should be understandable to the students who attended my recent RTP course.

May 30

Wed

Ruben Sanchez (Sheffield)

Topology Seminar

16:00

Computing Borel's regulator II

Hicks Seminar Room J11

Oct 2

Tue

Eugenia Cheng (Sheffield)

Topology Seminar

14:00

An operadic approach to $n$-categories

Hicks Seminar Room J11

Abstract:
Operads provide a way of studying loop spaces, by giving a formalism for keeping track of weakly associative multiplication. In this talk I will discuss how this is related to study of weak $n$-categories, where now we must keep track of weakly associative composition. I will present the definition of weak $n$-category proposed by Trimble, which uses one specific and very straightforward topological operad. This can be generalised so that we can use other operads such as the little intervals operad and possibly many of your favourite loop space operads.

Oct 9

Tue

Paul Mitchener (Sheffield)

Topology Seminar

14:00

Coarse Geometry

Hicks Seminar Room J11

Abstract:
Topology arises from the study of continuous maps, and essentially what happens at very small distances. Coarse geometry, by contrast, ignores all local structure, and only examines very large scale details. Essentially, all that matters in coarse geometry is what is going on `at infinity'. In this talk we will introduce the basic notions of coarse geometry, along with a number of examples and coarse invariants that are analogous to standard invariants in algebraic topology.

Oct 16

Tue

Teimuraz Pirashvili (Leicester)

Topology Seminar

14:00

Second Hochschild cohomology and triangulated categories

Hicks Seminar Room J11

Oct 30

Tue

Shoham Shamir (Sheffield)

Topology Seminar

14:00

Cellular approximations and the Eilenberg-Moore spectral sequence

Hicks Seminar Room J11

Abstract:
Given chain-complexes k and M over a ring R, a k-cellular approximation to M is the "closest approximation" of M that can be glued together from copies of suspensions of k. I will discuss this concept (due to Dwyer, Greenlees and Iyengar) and how is can be used to study the Eilenberg-Moore cohomology spectral sequence for a fibration.

Nov 6

Tue

James Cranch (Sheffield)

Topology Seminar

14:00

Spannish for beginners

Hicks Seminar Room J11

Abstract:
I will say something about the notion of a span category, the appropriate analogue in the language of quasicategories, and what all this is supposed to have to do with homotopy theory.

Nov 20

Tue

Christian Ausoni (Bonn)

Topology Seminar

14:00

On rational algebraic K-theory

Hicks Seminar Room J11

Abstract:
I will present a strategy for computing the rational algebraic K-theory of connective S-algebras. I will illustrate it in the cases where the algebra is connective complex or real topological K-theory. This is joint work with John Rognes (Oslo).

Nov 27

Tue

David Barnes (Sheffield)

Topology Seminar

14:00

Rational Equivariant Cohomology Theories

Hicks Seminar Room J11

Abstract:
If one wants to study spaces, one can use cohomology theories. For spaces with a group action, one uses equivariant cohomology theories which provide more refined information about the group action. By requiring that these cohomology theories are rational, one can study the collection of rational equivariant cohomology theories as a whole. In the case of a finite group, SO(2) or O(2) one can replace the collection of rational equivariant cohomology theories by an explicit and easy to understand algebraic category. I will talk about how, according to the group structure, the collection of rational equivariant cohomology theories splits into several disjoint collections. Thus one can study each of these pieces separately. I will also discuss how one can relate rational O(2) cohomology theories to rational SO(2) cohomology theories via the notion of a category with involution. This work is an overview of my thesis, supervised by John Greenlees.

Dec 11

Tue

Tony Hignett (Sheffield)

Topology Seminar

14:00

Discrete module categories

Hicks Seminar Room J11

Abstract:
A module over a topological ring is `discrete' if it is continuous when given the discrete topology. This concept is closely related to the coalgebra-algebra duality and hence to the cooperations-operations duality for a (decent) (co)homology theory E. I will talk about discrete module categories in general and the case E = K.

Jan 15

Tue

Bob Bruner (Wayne State)

Topology Seminar

14:00

Higher Leibniz Formulas

Hicks Seminar Room J11

Abstract:
The Leibniz formula tells us how differentials behave on products. When considering an S-algebra, there are higher order operations (Dyer-Lashof operations and their generalizations) and it is possible to work out formulas for differentials on these. They have been worked out in detail in two important cases, the Adams spectral sequence and the spectral sequence(s) for the homology of the homotopy fixed points, orbits or Tate construction of an $S^1$ equivariant S-algebra. In both cases, they provide a great deal of information about the differentials and extensions in the spectral sequence.

Jan 29

Tue

Wajid Mannan (Sheffield)

Topology Seminar

14:00

The dimension 2 problem

Hicks Seminar Room J11

Abstract:
This problem is an example of a phenomena which has long been known to hold in sufficiently high dimensions but is not known to hold in all low dimensions (in this case dimension 2). For n not equal to two, a finite cell complex of cohomological dimension n is homotopy equivalent to an n-complex. It is unknown whether this holds when n=2. I will discuss the problem and explain what I have done so far (proving that it holds sometimes) and mention what I am doing now (Vic's idea for finding a counterexample).

Feb 5

Tue

Johann Sigurdsson (Sheffield)

Topology Seminar

14:00

Homotopy operations

Hicks Seminar Room J11

Abstract:
I'll give a leisurely introduction to the theory of homotopy operations on categories of ring spectra.

Feb 12

Tue

Tom Bridgeland (Sheffield)

Topology Seminar

14:00

Wall-crossing and holomorphic generating functions

Hicks Seminar Room J11

Abstract:
To get nice moduli spaces for objects in algebraic geometry (e.g. vector bundles) one first has to choose a stability condition. As one varies this stability condition the moduli space of stable objects undergoes discontinuous changes. This is called wall-crossing behaviour. I will explain how this works in a simple example and describe some recent work of Joyce which allows one to make holomorphic generating functions for invariants associated to the moduli spaces using special functions related to multilogarithms.

Feb 19

Tue

Dirk Schuetz (Durham)

Topology Seminar

14:00

Cohomology of planar polygon spaces

Hicks Seminar Room J11

Abstract:
We study the topology of the moduli space of polygonal planar curves with given side-length vector. By a conjecture of Walker the side-lengths are determined by the cohomology ring of the moduli space. We show that this conjecture is true for a large class of length vectors, and that an analogous conjecture holds if one considers polygonal curves in 3-space. This is joint work with Michael Farber and Jean-Claude Hausmann.

Feb 26

Tue

Constanze Roitzheim (Sheffield)

Topology Seminar

14:00

Morita theory in stable homotopy theory

Hicks Seminar Room J11

Abstract:
In classical Morita theory, one uses the endomorphisms of a ring R to study the derived category of R-modules. We see how this generalises to studying the homotopy category of a stable model category by endomorphism ring specra. Further, we look at how Morita theory might help us classify algebraic models of the K-local stable homotopy category at odd primes.

Mar 4

Tue

Andrey Lazarev (Leicester)

Topology Seminar

15:30

Hicks Seminar Room J11

Apr 8

Tue

Sarah Whitehouse (Sheffield)

Topology Seminar

14:00

Robinson's bicomplex and Taylor towers

Hicks Seminar Room J11

Abstract:
Robinson's bicomplex was introduced to provide an obstruction theory for E-infinity structures on ring spectra. For suitable functors taking values in an abelian category, one can define a Taylor tower approximating the functor. In this expository talk, I will explain the relationship between Robinson's bicomplex and Taylor towers, namely the bicomplex is a model for the first layer of the tower. I will discuss recent work of Intermont-Johnson-McCarthy interpreting the rank filtration of functors in terms of the Robinson complex.

Apr 15

Tue

Paul Mitchener (Sheffield)

Topology Seminar

14:00

What is the Baum-Connes conjecture and why should we care?

Hicks Seminar Room J11

Abstract:
This talk should be a fairly gentle introduction to the formulation of the Baum-Connes conjecture, some generalisations and analogues, and topological implications of the conjecture, such as the Novikov conjecture, and the question of the existence of positive scalar curvature metrics on certain manifolds.

Apr 22

Tue

John Hunton (Leicester)

Topology Seminar

14:00

Cohomology of spaces of substitution tilings

Hicks Seminar Room J11

Abstract:
One of the main tools that have proved effective in studying aperiodic tilings has been the algebraic topology (cohomology or K-theory) of an associated moduli space of tilings locally equivalent to the individual tiling considered. A special class of examples are the tilings generated by substitutions, and although these are far from being generic examples, they include most of the well known and historically early examples (Fibonacci, Thue-Morse, Penrose, Amman-Beenker, etc). This talk will describe new techniques for understanding the cohomology of their associated spaces.

Apr 28

Mon

Morten Brun (Bergen)

Topology Seminar

14:00

Covering Homology

Hicks Seminar Room J11

Abstract:
Given a topological space $X$ and an abelian group $A$ there is a free topological abelian group $A \otimes X$ which morally it is the $X$-fold sum of copies of $A$. The homotopy of the undlying space of this topological abelian group is the homology of $X$ with coefficients in $A$. This approach to homology also works in other contexts. For example, if $A$ is a commutative ring then the commutative ring $A \otimes S^1$ is version of Hochschild homology. In the talk I shall focus on the situation where $A$ is a commutative ring-spectrum, and I shall explain how covering projections $X \to Y$ allow us to use this construction to obtain variations of Bökstedt, Hsiang and Madsen's topological cyclic homology.

Apr 29

Tue

Mark Grant (Durham)

Topology Seminar

14:00

Topological aspects of motion planning

Hicks Seminar Room J11

Abstract:
Inspired by the motion planning problem in robotics, M. Farber recently introduced a new numerical homotopy invariant, called the Topological Complexity, which provides a measure of the navigational complexity of a space when viewed as the configuration space of a mechanical system. As well as its practical motivation, computation of this invariant presents a challenge to homotopy theorists, which may be likened to computation of the Lusternik-Schnirelmann category. I will survey the best known lower and upper bounds for Topological Complexity, using plenty of examples. I also hope to discuss a recently obtained upper bound based on the homology coalgebra structure of the space.

Apr 29

Tue

Morten Brun (Bergen)

Topology Seminar

16:10

Equivariant multilinearity in algebra and topology

Hicks Seminar Room J11

Abstract:
The ring of (big) Witt vectors over a commutative ring appears naturally in the description of certain algebraic K-theory groups. These K-groups are related to equivariant stable homotopy via the topological Hochschild homology construction. It has been known for twenty years, that that given a (pro-)finite group G there is a G-typical version of the ring of Witt vectors. This G-typical Witt ring is related to commutative G-ring spectra, that is, commutative monoids in the G-equivariant stable homotopy category.

In the talk I will propose a generalization of the concept of multilinearity that gives a new approach to both Witt vector constructions and certain G-equivariant stable homotopy groups. In particular it can be used to describe the lowest homotopy group of G-fold smash-powers of G-spectra.

May 6

Tue

Jeff Giansiracusa (Oxford)

Topology Seminar

14:00

Pontrjagin-Thom maps and the Deligne-Mumford compactification.

Hicks Seminar Room J11

Abstract:
This is joint work with Johannes Ebert. We extend the classical construction of Pontrjagin-Thom wrong way maps to the setting of topological stacks. This construction applied to the boundary divisors of the Deligne-Mumford compactification produces many new mod p cohomology classes.

May 13

Tue

Simon Willerton (Sheffield)

Topology Seminar

14:00

The cardinality of a metric space.

Hicks Seminar Room J11

Abstract:
Hadwiger's Theorem says for a finite union of convex subsets in some Euclidean space that the Euler characteristic, perimeter, and so on up to the volume, are the only `additive', `invariant' measures. Note that lots of interesting spaces such as spheres and fractals are not finite unions of convex sets. The aim of the talk is to describe one way of trying to look at such measures on more general spaces. Tom Leinster defined the notion of Euler characteristic for a subclass of finite categories and has extended this idea to finite metric spaces by considering them as a certain type of enriched category. I will explain a conjectural connection with Hadwiger measures.

May 20

Tue

Kirill Mackenzie (Sheffield)

Topology Seminar

14:00

Lie Theory for Multiple Structures

Hicks Seminar Room J11

Abstract:
I plan to do two, perhaps three, main things in this talk: -- describe the Lie theory of (ordinary) Lie groupoids and its relation to connection theory; -- describe how Poisson group theory leads to multiple Lie structures; -- outline the Lie theory of double Lie groupoids. This will be an overview, not a technical talk. I'll recall notions from Poisson geometry and connection theory.

May 27

Tue

Richard Hepworth (Sheffield)

Topology Seminar

14:00

Orbifold Morse Homology

Hicks Seminar Room J11

Abstract:
Morse theory is a geometric way to understand the homology of manifolds. Orbifolds are spaces that locally look like the quotient of a manifold by a finite group. I will explain how Morse theory generalizes to orbifolds, giving methods to compute several different notions of "the homology of an orbifold" using generalizations of the Witten Complex.

Jun 3

Tue

Ieke Moerdijk (Sheffield)

Topology Seminar

14:00

A Milnor-Moore Theorem for Lie-Rinehart algebras

Hicks Seminar Room J11

Abstract:
Lie-Rinehart algebras arise naturally as the algebraic counterpart of Lie algebroids(which are the infinitesimal structures related to Lie groupoids). I will discuss to what extent the enveloping algebra of a Lie-Rinehart algebra carries a structure like that of a Hopf algebra, and discuss a Milnor-Moore type theorem for these structures.(The talk is based on a joint paper with J. Mrcun, available on the ArXiv.)

Jun 11

Wed

John Greenlees (Sheffield)

Topology Seminar

14:00

Hicks Seminar Room J11

Jun 26

Thu

Sharon Hollander (Lisbon)

Topology Seminar

14:00

Applications of Homotopy Theory of Stacks

Hicks Seminar Room J11

Abstract:
I will describe the homotopy theory of stacks and explain how algebraic stacks can be naturaly seen in this context. A consequence of this perspective will be certain criteria for the algebraicity of a stack.

Sep 30

Tue

Paul Mitchener (Sheffield)

Topology Seminar

14:00

Coarse Homotopy Theory

Hicks Seminar Room J11

Abstract:
The category of metric spaces and coarse maps does not carry the structure of a Quillen model category in any obvious way, for the simple reason that we do not know how to form products in the coarse category.

However, the coarse category can be equipped with a weaker structure- that of a Baues cofibration category. We show how to do this in this talk.

The cofibration category structure gives us an abstract notion of coarse homotopy groups. This abstract notion is closely related to something more geometric- the plan is to define this ``something'' in the talk and compute some simple examples.

Oct 7

Tue

Richard Hepworth (Sheffield)

Topology Seminar

14:00

2-Vector Bundles and Differentiable Stacks

Hicks Seminar Room J11

Abstract:
This seminar is an account of Alan Weinstein's recent paper The Volume of a Differentiable Stack. I'll explain that differentiable stacks are a generalization of smooth manifolds and that they crop up in many interesting situations, like the study of of orbifolds or the study of flat connections. Just as every manifold has a tangent bundle, every stack has a tangent something, and I'll explain that the something in question is a bundle of Baez-Crans 2-vector spaces. These 2-vector bundles are often horrible compared with vector bundles, but they still admit a 'top exterior power'. We'll see that sections of this top exterior power can be treated just like volume forms on a manifold, and in particular can be integrated to define the volume of a stack.

Oct 14

Tue

Neil Strickland (Sheffield)

Topology Seminar

14:00

Rational spectra and chain complexes

Hicks Seminar Room J11

Abstract:
In stable homotopy theory we study spectra with various kinds of structure, such as group actions or ring structures. Often it is illuminating to restrict attention to spectra whose homotopy groups are rational vector spaces, as many things become simpler and more algebraic in that context. Indeed, rational spectra without extra structure are essentially the same as rational chain complexes. The word 'essentially' hides some subtleties, which previously made it difficult to incorporate extra structures in the picture. I will report on a way to resolve this difficulty, which makes contact with de Rham theory in an unexpected way.

Oct 21

Tue

Hadi Zare (Manchester)

Topology Seminar

14:00

On spherical classes in $H_{\ast}QS^1$.

Hicks Seminar Room J11

Abstract:
This talk is about spherical classes in $H_{\ast}QS^1$. Inspired by work of Curtis and Wellington, we give an upper bound on the type of classes in $H_{\ast}QX$ which can be spherical. Specialising to $X=S^1$, the results can be refined. I will explain the motivation for studying this problem, and recall some results about this.

Oct 28

Tue

Kathryn Hess (Lausanne)

Topology Seminar

14:00

Power maps in algebra and topology

Hicks Seminar Room J11

Abstract:
(Joint work with J. Rognes) Let $t:C\to A$ be a twisting cochain, where $C$ is a connected, coaugmented chain coalgebra and $A$ is an augmented chain algebra over an arbitrary PID $R$. I'll explain the construction of a twisted extension of chain complexes $$A\to H(t)\to C$$ of which both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases. We call $H(t)$ the Hochschild complex of $t$. When $A$ is a chain Hopf algebra, I'll give conditions under which $H(t)$ admits an $r^{\text{th}}$-power map extending the usual $r^{\text{th}}$-power map on $A$ and lifting the identity on $C$. In particular, both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps. Moreover, if $K$ is a double suspension, then the power map on the coHochschild complex of the normalized chain complex of $K$ is a model for the topological power map on the free loops on $K$, illustrating the topological relevance of our algebraic construction. This algebraic model of the topological power map is a crucial element of the construction of our model for computing spectrum homology of topological cyclic homology of spaces.

Nov 4

Tue

Shoham Shamir (Sheffield)

Topology Seminar

14:00

Loops on a p-complete space and hereditary torsion theories

Hicks Seminar Room J11

Abstract:
Benson constructs a purely algebraic model for $H^*(\Omega (BG^\wedge_p);p)$, where $G$ is a finite group, $(-)^\wedge_p$ denotes the Bousfield-Kan $p$-completion. This construction can be generalized for the classifying space of any discrete monoid $M$, as long as $M$ is "nice". This gives an excuse to present some algebra, since Benson's construction uses the old algebraic notion of a hereditary torsion theory to calculate a certain localization functor on the derived category of $k[M]$, where $k$ is the field of $p$-elements. I will explain these notions, why they are interesting, and present the construction.

Nov 11

Tue

Ian Leary (Ohio and Bristol)

Topology Seminar

14:00

New Smith groups and Kropholler's hierarchy

Hicks Seminar Room J11

Abstract:
We construct an infinite group that has a very strong fixed point property for actions on finite-dimensional contractible spaces. Using similar techniques we show that Kropholler's hierarchy of groups is as long as it possibly could be: previously only the first four levels of the hierarchy were known to contain groups.

Nov 19

Wed

Kiyoshi Igusa (Brandeis)

Topology Seminar

14:00

Higher Reidemeister Torsion I:\\ Sphere Bundles

Hicks Seminar Room J11

Abstract:
Higher Reidemeister torsion can be defined using Morse theory (Igusa- Klein approach), homotopy theory (Dwyer-Weiss-Williams and Dorabiala) and analytically (Bismut-Lott and Goette). It is a challenge to see if these are equivalent. These talks are aimed at relating the Morse theory and homotopy theory points of view. The object of study is a smooth fiber bundle: $$ M\to E\to B $$ where $M,E,B$ are all compact smooth manifolds and the action of $ \pi_1B$ on the rational homology of $M$ is trivial. In this case all three invariants are defined. The easiest example is and oriented sphere bundle. 1) Sphere bundles By classical results about Euclidean bundles, topological sphere bundles have well-defined rational Pontrjagin classes. Smooth oriented sphere bundles have higher Reidemeister torsion invariants which are proportional to the topological Pontrjagin character for linear sphere bundles and for all even dimensional sphere bundles. When the fiber is an odd dimensional sphere, these invariants can differ and the difference measures the exotic smooth structure on the sphere bundles. I will discuss the theory of these exotic structures using Morse theory and the Dwyer-Weiss-Williams homotopy theoretic calculation of the group of fiberwise stable smooth structures on smooth bundles. I will also discuss the recent results of S. Goette comparing higher analytic torsion and the Morse theory version (IK-torsion) and the results of Goette and myself comparing IK-torsion and DWW-torsion.

Nov 20

Thu

Kiyoshi Igusa (Brandeis)

Topology Seminar

14:00

Higher Reidemeister Torsion II:\\ Dwyer-Weiss-Williams higher torsion and a construction of Hatcher.

Hicks Seminar Room J11

Abstract:
Higher Reidemeister torsion can be defined using Morse theory (Igusa- Klein approach), homotopy theory (Dwyer-Weiss-Williams and Dorabiala) and analytically (Bismut-Lott and Goette). It is a challenge to see if these are equivalent. These talks are aimed at relating the Morse theory and homotopy theory points of view. The object of study is a smooth fiber bundle: $$ M\to E\to B $$ where $M,E,B$ are all compact smooth manifolds and the action of $ \pi_1B$ on the rational homology of $M$ is trivial. In this case all three invariants are defined. 2) Dwyer-Weiss-Williams higher torsion and a construction of Hatcher In the second talk I will give my version of the Dwyer-Weiss-Williams theory of higher torsion. Basically, they show that the stable smooth structures on a topological manifold bundle $E\to B$ with prescribed vertical tangent bundle are classified by sections of the associated $H^{%}$-bundle (the bundle over $B$ whose fibers are $\Omega^ \infty(M_+\wedge \mathcal H(\ast))$ where $\mathcal H(X)$ is the stable smooth concordance space of $X$. In the case when $M,E,B$ are all closed manifolds, this is given rationally by a homology class in $E$ which we call the stable smooth structure class. The Poincaré dual of the image of this class in the homology of $B$ is the higher DWW-torsion. Using a generalization of a construction of Hatcher, Goette and I constructed sufficiently many exotic smooth structures on any bundle and calculated their IK-torsion and we concluded that IK- torsion and DWW-torsion agree up to a constant. (However, this is not the complete answer since we prescribed the vertical tangent bundle.)

Nov 21

Fri

Kiyoshi Igusa (Brandeis)

Topology Seminar

14:00

Higher Reidemeister Torsion III:\\ Iterated integrals, superconnections and higher torsion

Hicks Seminar Room J11

Abstract:
Higher Reidemeister torsion can be defined using Morse theory (Igusa- Klein approach), homotopy theory (Dwyer-Weiss-Williams and Dorabiala) and analytically (Bismut-Lott and Goette). It is a challenge to see if these are equivalent. These talks are aimed at relating the Morse theory and homotopy theory points of view. The object of study is a smooth fiber bundle: $$ M\to E\to B $$ where $M,E,B$ are all compact smooth manifolds and the action of $ \pi_1B$ on the rational homology of $M$ is trivial. In this case all three invariants are defined. The easiest example is and oriented sphere bundle. 3) Iterated integrals, superconnections and higher torsion This talk explains how iterated integrals are used in the definition and calculation of higher torsion. Given a smooth fiber bundle, we can construct an $A_\infty$-functor from the category of smooth simplices in the base to the $A_\infty$-category of finitely generated chain complexes over a field. Taking the limit as the size of the simplices go to zero we get a flat $\mathbb Z$-graded superconnection on the base. Conversely, such a superconnection can be integrated using Chen's iterated integrals to recover the $A_\infty$-functor. The higher Reidemeister torsion can be defined categorically using the $A_ \infty$-functor. However, to calculate it one needs an explicit formula for the flat superconnection. I will talk about the relation between these three concepts.

Nov 25

Tue

Nick Wright (Southampton)

Topology Seminar

14:00

Property A and dimension for CAT(0) cube complexes.

Hicks Seminar Room J11

Abstract:
Yu's property A is a property of a space which is a geometric analogue of amenability for groups. I will present a result on property A for CAT(0) cube complexes, and discuss strengthening this result in terms of the large-scale dimension of these spaces. These questions are motivated in part by open questions about Thompson's group F.

Dec 2

Tue

Bruce Bartlett (Sheffield)

Topology Seminar

14:00

Pivotal structures on fusion categories

Hicks Seminar Room J11

Abstract:
A fusion category is a monoidal category whose hom-sets are finite-dimensional vector spaces and which is semisimple --- in the sense that there are a finite bunch of 'simple' objects, and every other object is a direct sum of them. Fusion categories arise in several areas of mathematics and physics: conformal field theory, operator algebras, representations of quantum groups, and so on. A conjecture was made by Etingof, Nikshych and Ostrik that ''every fusion category admits a pivotal structure''. In this talk I will explain what that means, and I will present some work which might help in settling this conjecture. Specifically, I will use a string diagram argument first discovered by Hagge and Hong, similar to the Dirac belt trick, which shows that the hom-sets in a fusion category carry involution operators, which must be "trivial" in order for the category to admit a pivotal structure.

Feb 5

Thu

Eugenia Cheng (Sheffield)

Topology Seminar

15:05

Ubiquitous Yoneda: universal operads

Hicks Seminar Room J11

Abstract:
We show how to build operads for n-categories that are informally analogous to the universal loop space operads. We will make this universal property precise by showing that it is in fact the Yoneda Lemma in disguise. We will then explain how this constitutes a win in the Australian version of "Mornington Crescent".

Feb 12

Thu

John Jones (Warwick)

Topology Seminar

15:05

Batalin Vilkovisky algebras and string homology

Hicks Seminar Room J11

Abstract:
String homology was introduced by Moira Chas and Dennis Sullivan in 1999. Their idea was to do intersection theory on the loop space of a finite dimensional manifold. In a subsequent paper, published in 2002, Ralph Cohen and myself gave a different approach to the theory using the general methods of algebraic topology and homotopy theory. One of the outputs of string homology is that the theory shows how to associate an algebraic structure known as a Batalin Vilkovisky algebra to a closed finite dimensional manifold. In this talk I will discuss Batalin Vilkovisky algebras and how they arise in algebraic topology, in particular in string homology, and emphasize two fundamental problems. 1. How does one calculate string homology? 2. What exactly does string homology depend on?

Feb 19

Thu

Andrew Stacey (Trondheim)

Topology Seminar

15:05

Comparative Smootheology

Hicks Seminar Room J11

Abstract:
"Manifolds are lovely spaces; it's just a pity there aren't enough of them." In my work on loop spaces I have often come across the problem that loop spaces are like ordinary manifolds but not completely alike. One has to be careful when taking ideas and techniques from ordinary differential topology and geometry to spaces like loop spaces. Considerations like this have led a variety of researchers to propose notions of "generalised smooth spaces". Unfortunately, there are a lot of these notions about. In this talk I shall explain why I like "Frolicher spaces" best of all the different versions. I shall also comment a little on other topics, in particular the differences and similarities between the various notions.

Feb 26

Thu

Simon Willerton (Sheffield)

Topology Seminar

15:05

Two 2-traces

Hicks Seminar Room J11

Abstract:
Monoidal bicategories are not scary abstract beasts but crop up concretely in many places in algebra and topology; I will use several examples as the backbone to the talk. In a monoidal bicategory there are two different notions of trace for endomorphisms which in various cases are `dual'. I will illustrate with various pictures and examples.

Mar 5

Thu

Harry Ullman (Sheffield)

Topology Seminar

15:05

Equivariant generalizations of Millers stable splitting

Hicks Seminar Room J11

Abstract:
In 1985 Miller proved that Stiefel manifolds, and in particular the unitary group, split stably as a wedge of Thom spaces over Grassmannians. This talk will discuss efforts towards generalizing Miller's results in an equivariant setting including a main conjecture, a survey of results found so far and an explanation as to just why putting $G$ in front of everything in sight isn't the right thing to do.

Mar 9

Mon

Dev Sinha (Oregon)

Topology Seminar

16:10

Cohomology of symmetric groups

Hicks Seminar Room J11

Mar 10

Tue

Dev Sinha (Oregon)

Topology Seminar

13:10

Hopf invariants

Hicks Seminar Room J11

Mar 12

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

15:05

Infinity Categories and Infinity Operads I

Hicks Seminar Room J11

Mar 16

Mon

Ieke Moerdijk (Sheffield)

Topology Seminar

16:00

Infinity Categories and Infinity Operads II

Hicks Seminar Room J11

Mar 19

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

15:05

Infinity Categories and Infinity Operads III

Hicks Seminar Room J11

Mar 26

Thu

Constanze Roitzheim (Glasgow)

Topology Seminar

15:05

Hochschild cohomology of A-infinity algebras

Hicks Seminar Room J11

Abstract:
In the 1960s, A-infinity algebras were introduced to study the cohomology of topological spaces with products and are now known to arise widely in various areas of mathematics. Roughly speaking, A-infinity algebras are generalisations of associative algebras. We are going to explain how to extend the definition of Hochschild cohomology from associative algebras to A-infinity algebras and how this will help solving realizability problems in topology.

Apr 2

Thu

Elizabeth Hanbury (Durham)

Topology Seminar

15:05

Simplicial structures on braid groups and mapping class groups

Hicks Seminar Room J11

Apr 30

Thu

Kijti Rodtes (Sheffield)

Topology Seminar

15:05

The connective $k$ theory of a semidihedral group

Hicks Seminar Room J11

Abstract:
For a finite group G, $ko_*(BG)$ plays a role in Gromov-Lawson-Rosenberg conjecture. We can compute it via $ku^*(BG)$ by using Bockstein spectral sequence and Greenlees spectral sequence. In this talk, we will show how to calculate $ku^*(BG)$ and $ku_*(BG)$ where $G$ is the semidihedral group of order 16.

May 7

Thu

Hao Zhao (Manchester)

Topology Seminar

15:05

Homotopy exponents of some homogeneous spaces

Hicks Seminar Room J11

Abstract:
Let p be a prime. Using the methods of homotopy decomposition and spherical fibrations, under some restricted conditions we obtain upper bounds for the $p$-primary homotopy exponents of some homogeneous spaces such as the complex Stiefel manifold, complex Grassmann manifold, $SU(2n)/Sp(n)$, $E_{6}/F_{4}$ and $F_{4}/G_{2}$.

May 14

Thu

Assaf Libman (Aberdeen)

Topology Seminar

11:30

The gluing problem and Bredon cohomology

Hicks Seminar Room J11

May 14

Thu

Andras Juhasz (Cambridge)

Topology Seminar

14:30

Classifying minimal genus Seifert surfaces

Hicks Seminar Room J11

Abstract:
First I will survey two different notions of equivalence for Seifert surfaces. Then I will show how sutured Floer homology helps in the classification of minimal genus Seifert surfaces under both types of equivalence.

May 14

Thu

Michael Farber (Durham)

Topology Seminar

16:15

Topology of random manifolds

Hicks Seminar Room J11

Abstract:
Betti numbers of configuration spaces of mechanical linkages (known also as polygon spaces) depend on a large number of parameters -- the lengths of the bars of the linkage. Motivated by applications in topological robotics, statistical shape theory and molecular biology, we view these lengths as random variables and study asymptotic values of the average Betti numbers as the number of links n tends to infinity. We establish a surprising fact that for a reasonably ample class of sequences of probability measures the asymptotic values of the average Betti numbers are independent of the choice of the measure.

May 28

Thu

Professor Rick Jardine (University of Western Ontario)

Topology Seminar

15:10

Pointed torsors and Galois groups

Hicks Seminar Room J11

Abstract:
Suppose that H is an algebraic group which is defined over a field k, and let L be the algebraic closure of k. The canonical stalk for the etale topology on k induces a simplicial set map from the classifying space B(H-tors) of the groupoid of H-torsors (aka. principal H-bundles) to the space BH(L). The homotopy fibres of this map are groupoids of pointed torsors, suitably defined. These fibres can be analyzed with cocycle techniques: their path components are representations of the absolute Galois groupoid of k in H, and each path component is contractible. The arguments for these results are simple, and applications will be displayed.

Jun 4

Thu

Nick Kuhn (Virginia)

Topology Seminar

15:05

Detection numbers in group cohomology

Hicks Seminar Room J11

Abstract:
Let $H^\ast(BG)$ denote the mod p cohomology of the classifying space of a compact Lie group G (e.g. a finite group). Since Quillen's work around 1970, $H^\ast(BG)$ has been fruitfully studied via restriction to its various elementary abelian p--subgroups $V$. In the early 1990's, Henn, Lannes, and Schwartz generalized Quillen's work. In particular, they define $d_0(G)$ as the smallest d such that the evident restriction map $$H^\ast(BG)\to\Pi_{V\le G} H^\ast(BG)\otimes H^{\ast\le d}(BC_G(V))$$ is monic. I will describe a way to calculate an upper bound for $d_0(G)$ using information that is often easy to compute before one knows much about $H^\ast(BG)$. The bound seems very good in general, and is exact for many groups, e.g. finite groups for which every element of order p is central in a p--Sylow subgroup. The story of why our bound works goes as follows. Firstly, our extensive knowl- edge of $H^\ast(BG)$ as an unstable module over the mod p Steenrod algebra leads us to the study of the primitives in the central essential cohomology of $BG$, viewed as a comodule over the cohomology of its maximal central elementary abelian p-- subgroup. Then we use Hopf algebra tricks, as in work of Duflot, Broto, Henn, and D. Green, to control these primitives. This allows us to connect our problem to properties of the local cohomology of $H^\ast(BG)$ as studied by Benson, Carlson, and Greenlees. Finally, a new theorem of Symonds, establishing Benson's Regularity Conjecture, tells us what we need. Examples will be given. For example, when $p = 2$, $d_0(SU(3, 4)) = 14$, and this is biggest among all finite groups having a 2--Sylow subgroup of order 64 or less.

Sep 29

Tue

Paul Mitchener (Sheffield)

Topology Seminar

14:00

General Descent

Hicks Seminar Room J11

Abstract:
The term "descent" in coarse geometry usually means the fact that the coarse Baum-Connes conjecture (plus certain mild extra conditions) implies injectivity of the assembly map in the ordinary Baum-Connes conjecture. All of this can be generalised to a general notion of assembly maps; there is a corresponding "coarse isomorphism conjecture", which implies that the assembly map is injective. Thus, coarse techniques can be used to prove injectivity of a variety of assembly maps.

Oct 6

Tue

John Greenlees (Sheffield)

Topology Seminar

13:30

Rational torus-equivariant cohomology theories.

Hicks Seminar Room J11

Abstract:
The talk will describe a model for these cohomology theories for a torus G, and potential applications. The algebraic model A(G) is an abelian category of injective dimension equal to the rank of G, based on the use of idempotents in Burnside rings and the Borel-Hsiang-Quillen localization theorem for passage to torus- fixed points. Its formal structure is rather like that of structured sheaves over an r dimensional variety (this, naturally, guides some of the applications, such as cohomology theories associated to higher dimensional abelian varieties). The talk may describe the strategy of proof in joint work with Shipley, based on rigidity and building up data through an isotropic Hasse-square.

Oct 13

Tue

Shoham Shamir (Sheffield)

Topology Seminar

13:30

Complete intersections in rational homotopy theory

Hicks Seminar Room J11

Abstract:
In commutative algebra, complete intersection rings are the next best thing after regular rings. The quotient of a graded polynomial ring by a regular ideal is a prime example of a complete intersection ring. Gulliksen showed that a local Noetherian ring is complete intersection if and only if its homology has polynomial growth. Benson and Greenlees recently characterized local complete intersection rings by the existence of a certain structure on their derived category. These definitions have obvious adaptations for rational spaces. For simply connected rational spaces these adapted definitions are shown to be equivalent, yielding a structural characterization of complete intersection rational spaces using spherical fibrations. This is joint work with John Greenlees and Kathryn Hess.

Oct 20

Tue

Carl McTague (Cambridge)

Topology Seminar

13:30

The Cayley Plane and the Witten Genus

Hicks Seminar Room J11

Abstract:
Elliptic cohomology is at the heart of many recent developments in algebraic topology. (Hill-Hopkins-Ravenel for example recently used it to solve the Kervaire invariant problem.) What led to its discovery was Ochanine's observation in the 1980s that there are many more multiplicative genera for spin fiber bundles than for oriented fiber bundles, one for each elliptic curve with a marked point of order 2. Given that multiplicative genera for spin fiber bundles have led to such unexpectedly rich developments, it seems reasonable to investigate multiplicative genera for other types of fiber bundles, in particular O<8> fiber bundles. I will discuss a recently published result of Dessai and a result of my own which in investigating this question place the Witten genus into a geometric framework.

Oct 27

Tue

Arjun Malhotra (Sheffield)

Topology Seminar

13:30

The Gromov-Lawson-Rosenberg conjecture for some finite groups

Hicks Seminar Room J11

Abstract:
The Gromov-Lawson-Rosenberg conjecture for a group G says that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. The plan is to discuss the conjecture, and sketch how to prove it for some finite groups.

Nov 10

Tue

Ieke Moerdijk (Sheffield)

Topology Seminar

13:30

Deformation theory of Lie algebroids, I

Hicks Seminar Room J11

Abstract:
The notion of Lie algebroid encompasses Lie algebras, foliations, infinitesimal actions, Poisson manifolds and other geometric structures. I will describe a differential graded Lie algebra which controls deformations of Lie algebroids. The corresponding deformation cohomology agree with the classical (Nijenhuis-Richardson) theory for Lie algebras, and captures some known results about deformations of foliations (Heitsch) and Poisson manifolds. The difficulty to overcome lies in the fact that there is no adjoint representation for Lie algebroids; in fact, one way to interpret our results is as the beginnings of a theory of representations-up-to-homotopy for Lie algebroids. (joint work with Crainic, reference: J. Eur. Math. Soc. 2008)

Nov 17

Tue

Andrew Baker (Glasgow)

Topology Seminar

13:30

$E_\infty$ ring spectra related to $BP$

Hicks Seminar Room J11

Abstract:
I will describe the construction of a commutative $S$-algebra which is tantalisingly close to the Brown-Peterson spectrum at the prime $2$. The ingredients are power operations and calculations using the Adams spectral sequence.

Nov 24

Tue

James Cranch (Leicester)

Topology Seminar

13:30

Pictures of Distributivity

Hicks Seminar Room J11

Abstract:
I'll talk a bit about algebraic theories: these are an approach to wrapping the axioms for many algebraic structures into a pleasant categorical package. I'll also say something about the higher-categorical version of algebraic theories which I introduced in my PhD thesis to study questions in topology. Then I'll describe what theories look like whose operations satisfy a distributive law (like the theory of rings, in which multiplication distributes over addition). There will be pictures and hopefully even a physical model of the 3D "distributahedron".

Dec 1

Tue

Nigel Ray (Manchester)

Topology Seminar

13:30

Realisations of the Stanley-Reisner algebra and homotopy uniqueness

Hicks Seminar Room J11

Abstract:
This talk will be a report on joint work with Dietrich Notbohm (VU Amsterdam). In 1991, for any finite simplicial complex K, Davis and Januszkiewicz defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. In 2002, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by such a cohomology ring. I shall outline our analysis of this problem i) rationally, and ii) prime by prime, and then attempt to explain how the outcomes may be reassembled using Sullivan's arithmetic square. The entire problem becomes straightforward after a single suspension, and I shall start by discussing this case as a warm-up exercise.

Dec 8

Tue

Simon Willerton (Sheffield)

Topology Seminar

13:30

The asymptotic magnitude of surfaces

Hicks Seminar Room J11

Jan 20

Wed

Urs Schreiber (Utrecht)

Topology Seminar

16:00

Path-structured oo-toposes

Hicks Seminar Room J11

Abstract:
The description of differential String-structures, a central ingredient in certain geometrically defined quantum field theories, requires a nonabelian generalization of differential generalized cohomology. This can be constructed in terms of smooth path $\infty$-groupoids of smooth $\infty$-stacks. I describe these and indicate how they give rise to Chern characters in deRham cohomology on $\infty$-stacks.

Jan 26

Tue

Urs Schrieber (Utrecht)

Topology Seminar

16:00

Path-structured $\infty$-toposes, part 2

Hicks Seminar Room J11

Feb 9

Tue

Nick Gurski (Sheffield)

Topology Seminar

15:00

Homotopy theory for 2-categories

Hicks Seminar Room J11

Abstract:
I will discuss a general technique for getting a model category structure (in fact, a Cat-enriched model category structure) on a 2-category. The weak equivalences will be the internal equivalences in your 2-category, and the fibrations will be the internal isofibrations. Both of these kinds of morphisms are quite easy to define, and proving the model category axioms requires using some very basic 2-dimensional limits and colimits. Given time, I will say something about how one can then lift these model structures to produce some much more interesting examples.

Feb 23

Tue

Dirk Schuetz (Durham)

Topology Seminar

15:00

Sigma invariants, finiteness properties and closed 1-forms

Hicks Seminar Room J11

Abstract:
Sigma invariants, defined by Bieri-Neumann-Strebel-Renz, of a group G capture, among other things, finiteness properties of kernels of homomorphisms of G into the reals. As with finiteness properties, there exist homological and homotopical versions of these invariants, and due to the groundbreaking work of Bestvina and Brady it is known that they are different in general. We further investigate the differences between homological and homotopical invariants and study its impact on the existence of nonsingular closed 1-forms on closed manifolds of high dimension.

Mar 2

Tue

Neil Strickland (Sheffield)

Topology Seminar

15:00

Tambara functors

Hicks Seminar Room J11

Abstract:
Let $R$ be a strictly commutative ring spectrum with an action of a finite group $G$; then the homotopy group $\pi_0(R)$ fits into an algebraic structure known as a Tambara functor. We will discuss the algebraic theory of Tambara functors and their relationship with Witt rings, which have a number of different applications in stable homotopy theory.

Mar 9

Tue

Michael Joachim (Muenster)

Topology Seminar

15:00

Equivariant cohomotopy for infinite discrete groups

Hicks Seminar Room J11

Mar 16

Tue

Eugenia Cheng (Sheffield)

Topology Seminar

15:00

Iterated distributive laws via the Gray tensor product

Hicks Seminar Room J11

Abstract:
Monads give us a way of expressing algebraic structure, and distributive laws between monads give us a way of combining two types of algebraic structure. The basic example combines the free monoid monad (for multiplication) and the free Abelian group monad (for addition) via the usual distributive law, giving us the free ring monad. We give a framework for combining $n$ monads on the same category via distributive laws satisfying Yang-Baxter equations, showing that this way of distributing algebraic structure behaves somewhat like braids.

While it is possible to prove this using a very dull induction, one might wonder why on earth the Yang-Baxter equations popped up here. So I prefer to present a proof that emphasises the geometry of the situation, using the Gray tensor product for 2-categories.

Apr 13

Tue

Emmanuel Farjoun (Jerusalem)

Topology Seminar

15:05

Homotopy Normal maps of Monoids

Hicks Seminar Room J11

Apr 27

Tue

Gery Debongnie (Manchester)

Topology Seminar

15:00

On the rational homotopy type of subspace arrangements

Hicks Seminar Room J11

Abstract:
We shall explore different properties of the complement spaces of subspace arrangements, from the viewpoint of rational homotopy theory. A rational model will be described, from which we deduce several results. For example, we give a complete description of coordinate subspace arrangements whose complement space is a product of spheres.

May 11

Tue

Andrei Akhvlediani (Oxford)

Topology Seminar

15:00

On the categorical meaning of Gromov and Hausdorff distances.

Hicks Seminar Room J11

Abstract:
By interpreting the distance $d(x,y)$ as $\hom(x,y)$, Lawvere considered metric spaces as categories enriched in the extended positive reals. This viewpoint led to the adoption of tools of enriched category theory in the study of metric spaces; its usefulness is evident already in the work of Leinster and Willerton on the magnitude of metric spaces.

In this talk we will use enriched category theory to analyse the Gromov distance, which is a metric on the class of isometry classes of compact metric spaces, and its precursor - the Hausdorff metric. We exhibit the Hausdorff metric as part of a monad and define Gromov distance in terms of so-called $V$-modules. The categorical viewpoint allows us to pursue those distances in great generality and reveals some of their algebraic properties.

Some familiarity with category theory will be helpful, but not essential.

May 18

Tue

Kirill Mackenzie (Sheffield)

Topology Seminar

16:00

Lie bialgebroids

Hicks Seminar Room J11

Abstract:
Lie bialgebroids were introduced by the speaker and Xu Ping in 1994. This will be a very unhistorical tour d'horizon, with much benefit of hindsight.

A Poisson bracket on a manifold M is usually defined as an R-Lie algebra structure on the algebra of smooth functions, which is also a derivation in each variable. This induces a bracket on the 1-forms which behaves very much like the bracket of vector fields. These two bracket structures -- on TM and T^*M (or rather, on the modules of sections of TM and T^*M) -- resemble the situation in a Lie bialgebra.

Lie bialgebras arose in Drinfel'd's work in the 1980s, in part as semiclassical limits of quantum groups. There is now an extensive literature.

Lie bialgebroids were originally seen as a unifying concept, allowing Lie bialgebras and general Poisson manifolds to be treated simultaneously. They turned out to provide examples of differential Gerstenhaber algebras, Courant algebroids, and Dirac structures. In a different direction, they arise in the theory of double Lie groupoids.

A vague acquaintance with Poisson algebras or Poisson manifolds is desirable, though basics will be recalled. My notes from the Quantization seminar are

http://kchmackenzie.staff.shef.ac.uk/shef-only/poisson-09-10-26.pdf

and contain far more than is needed for this talk.

May 25

Tue

Richard Hepworth (Copenhagen)

Topology Seminar

15:05

Groups, Discs and Cacti

Hicks Seminar Room J11

Abstract:
The "framed little discs operad" is a topological gadget that acts on the double loop space of any based space X. The "cactus operad" is a gadget of the same kind, which this time (almost) acts on the free loops in a manifold M. The two operads are known to be homotopy equivalent. The purpose of the talk is to elaborate on the relationship between cacti and framed discs. First we will introduce a new action of cacti on the space of based loops in a topological group, and then we will show that it is equivalent to the action of framed discs on double loopspaces. Along the way we will give a new equivalence between cacti and framed discs.

Sep 28

Tue

Paul Mitchener (Sheffield)

Topology Seminar

13:45

KK-theory spectra and assembly

Hicks Seminar Room J11

Abstract:
The plan is to introduce assembly maps in various settings (including both algebraic and analytic K-theory) and general classification results involving assembly maps.

The analytic assembly map is classically defined in terms of KK-theory, and needs some work to express in a way where the classification machinery can be used. I will explain this process.

As time permits, I will also show how the homotopy algebraic K-theory assembly map, which is usually defined with the general machinery, can be expressed in terms of the bivariant algebraic KK-theory developed by Cortinas and Thom.

This is useful for computations.

Oct 5

Tue

Boris Botvinnik (Oregon)

Topology Seminar

14:00

The moduli space of generalized Morse functions

Hicks Seminar Room J11

Oct 7

Thu

Andrew Baker (Glasgow)

Topology Seminar

15:05

Galois theory for Lubin-Tate cochains on classifying spaces

Hicks Seminar Room J11

Abstract:
I'll discuss some results on Galois theory of the extension of Lubin-Tate cochain spectra $$E^{BG} = F(BG_+,E) \to F(EG_+,E) \equiv E,$$ where $E$ is a Lubin-Tate spectrum and $G$ is a finite group. In contrast to the case of $F(BG_+,HF_p)$, it turns out that this is always a faithful extension, but not always Galois.

Oct 12

Tue

Richard Hepworth (Copenhagen)

Topology Seminar

14:00

Higher categories and configuration spaces

Hicks Seminar Room J11

Abstract:
Joyal introduced categories $\Theta_n$ in order to define a theory of `weak n-categories'. These $\Theta_n$ also appear in Rezk's recent approach to the same question. This talk will report on joint work with David Ayala, where we show how the $\Theta_n$ encode combinatorial models for configuration spaces of points in $\mathbb{R}^n$. If time permits then I will describe some ambitions regarding Lurie's topological chiral homology.

Oct 19

Tue

Kijti Rodtes (Sheffield)

Topology Seminar

14:00

Real connective K theory of finite groups

Hicks Seminar Room J11

Abstract:
Real connective K theory of finite groups, $ko_{*}(BG)$, plays a big role in the Gromov-Lawson-Rosenberg (GLR) conjecture. To calculate it, we can proceed in several ways, e.g., by using the Atiyah-Hirzbruch spectral sequence, by the Adams spectral sequence or by the Greenlees spectral sequence (GSS). However, it is evident that the lattermost way, GSS, is very powerful and suitable for tackling the GLR conjecture. In this talk, we will show how to compute real connective K theory by using Bruner-Greenlees methods.

Oct 26

Tue

David Barnes (Sheffield)

Topology Seminar

14:00

Monoidality of Exotic Models

Hicks Seminar Room J11

Abstract:
The category of $K_{(p)}$-local spectra is an important approximation to the stable homotopy category that is somewhat easier to study. When p=2 this category is rigid, that is, all of the higher homotopy information of $K_{(2)}$-local spectra is contained in the triangulated structure of the homotopy category.

For $p=3$ this is not true, as well as $K_{(3)}$-local spectra there is the exotic model of Franke. The homotopy category of this exotic model has the same triangulated structure as $K_{(3)}$-local spectra, but arises from a different homotopy theory.

This talk will report on joint work with Constanze Roitzheim, where we show how to define a monoidal product for this exotic model, relate it to the smash product of $K_{(3)}$-local spectra and then compute the Picard group of the exotic model.

Nov 2

Tue

Sarah Whitehouse (Sheffield)

Topology Seminar

14:00

Central cohomology operations and $K$-theory

Hicks Seminar Room J11

Abstract:
In various contexts $K$-theory operations can be shown to map to operations of other cohomology theories, in such a way that the image of this map is precisely the centre of the target ring. I will discuss some results of this sort, both old and new, including joint work with Imma GÁlvez and M-J Strong.

Nov 9

Tue

Bob Bruner (Wayne State )

Topology Seminar

14:00

Ossa's theorem, Pic(A(1)) and generalizations

Hicks Seminar Room J11

Abstract:
Ossa's calculation of the complex connective K-theory of classifying spaces of elementary abelian groups depends upon the idempotence of a particular module over the exterior algebra on two generators. For the real connective K-theory, the algebra is more subtle. We give a particularly simple way to understand it, and relate the results to two localizations of the category of A(1)-modules and their Pic groups. I will end with comments and conjectures about higher analogs.

Nov 23

Tue

Eugenia Cheng (Sheffield)

Topology Seminar

14:00

Distributive laws for Lawvere theories

Hicks Seminar Room J11

Abstract:
Lawvere Theories and monads are two ways of handling algebraic theories. They are related but subtly different; one way in which they differ is that models for a given Lawvere Theory can automatically be taken in many different base categories, whereas monads have a fixed base category.

Distributive laws give a way of combining two algebraic structures expressed as monads, so one might naturally ask whether something analogous can be done for Lawvere Theories. In this talk I will give a way of doing this, using a reformulation of Lawvere Theories that is of interest in its own right. I will also discuss an illuminatingly wrong way of doing it.

I will not assume any prior knowledge of Lawvere Theories, so the first part of the talk will serve as an introduction to these things.

Nov 28

Sun

Jacob Rasmussen (Cambridge)

Topology Seminar

15:00

Hicks Seminar Room J11

Nov 28

Sun

Jacob Rasmussen (Cambridge)

Topology Seminar

15:00

Hicks Seminar Room J11

Nov 30

Tue

Nick Gurski (Sheffield)

Topology Seminar

14:00

Two-dimensional braids

Hicks Seminar Room J11

Abstract:
Braids occur naturally in topology, category theory, and other fields like representation theory, and the basic theory of the braid groups could be considered classical. On the other hand, "two-dimensional braids" are much newer objects of study that seem to arise from far more complicated algebra. In this talk I will introduce the study of two-dimensional braids using category theory, topology, and geometry, and will explain how the interactions between these various fields helps to show that the algebra of two-dimensional braids is actually simpler than it first appears.

Dec 7

Tue

John Hunton (Leicester )

Topology Seminar

14:00

What is an attractive shape?

Hicks Seminar Room J11

Abstract:
Suppose we have a differentiable manifold M with a self diffeomorphism yielding an expanding, hyperbolic attractor A. What can we say about the topology of A? In the case that A is of codimension 1 in M, we show that A can be modeled as a moduli space of an aperiodic tiliing and, conversely, we obtain conditions for when a tiling space can be embedded nicely in a manifold. These results give insights into the shape of such attractors, and new topological invariants for tilings, finer than the usual cohomological tools used in the subject.

Dec 14

Tue

Nige Ray (Manchester)

Topology Seminar

14:00

Toric methods in cobordism theory

Hicks Seminar Room J11

Abstract:
I shall recall certain basic aspects of real and complex cobordism theory, and explain how toric and quasitoric manifolds have enriched the theory since 1986, albeit unwittingly at first. I shall also describe a conjecture concerning stably $SU$-structures. Finally, I shall discuss the universal toric genus for equivariant cobordism, and consider its values on omnioriented quasitoric manifolds. Most of this work is joint with Victor Buchstaber and Taras Panov, or due to Alastair Darby.

Feb 15

Tue

Harry Ullman (Sheffield)

Topology Seminar

15:00

The equivariant stable homotopy theory of isometries

Hicks Seminar Room J11

Abstract:
Non-equivariantly, a space of linear isometries admits a stable splitting. In an equivariant setting, however, this does not generally happen. Instead, one can naturally build an equivariant stable tower with interesting topological properties similar to those exhibited by the non-equivariant splitting. We discuss this construction, while also mentioning obstructions to producing an equivariant splitting. Finally, we mention work-in-progress on retrieving a stable splitting from the tower in the special case where an equivariant splitting is possible.

Feb 22

Tue

Simon Willerton (Sheffield)

Topology Seminar

15:00

Hicks Seminar Room J11

Mar 1

Tue

Laura Stanley (Sheffield)

Topology Seminar

15:00

Upper Triangular Technology for odd primary K-Theory

Hicks Seminar Room J11

Abstract:
First published in 2002, Vic Snaith proved an isomorphism between a group of automorphisms of certain smash products of 2-complete connective K-Theory spectra and a group of infinite upper triangular matrices with entries in the 2-adic numbers. This would allow these infinite matrices to be used as a tool for studying maps of K-Theory spectra. Later, Snaith and his PhD student Jonathan Barker showed which matrix the Adams operation $\psi^3$ corresponds to under the isomorphism.

In this talk I will present the results of my thesis which are the corresponding odd primary analogues of both of these results, give an idea of how to prove them and indicate how the method generalises to tell us things about p-local K-Theory operations.

Mar 8

Tue

James Cranch (Leicester)

Topology Seminar

15:00

The structure of cofibre sequences

Hicks Seminar Room J11

Abstract:
I'll start by reminding people how the classical duality theorems for manifolds have evolved to follow various technological revolutions in algebraic topology, and then I'll speculate about how they might evolve in the near future. I'll explain how a modern understanding of Lefschetz duality -- the duality theory for manifolds with boundary -- would seem to require (among other things) an understanding of some interesting structure on cofibre sequences. Then I'll demonstrate what I've worked out about that structure.

Mar 15

Tue

Siu Por Lam

Topology Seminar

15:00

Equivariant K theory and equivariant Real K-theory of some spaces

Hicks Seminar Room J11

Mar 21

Mon

Philipp Wruck (Hamburg)

Topology Seminar

14:05

Geometrical Aspects of Topological Invariants

Hicks Seminar Room J11

Mar 22

Tue

Frank Neumann (Leicester)

Topology Seminar

15:00

Weil conjectures for the moduli stack of vector bundles on an algebraic curve

Hicks Seminar Room J11

Abstract:
In 1949 Weil conjectured deep connections between the topology and arithmetic of algebraic varieties over a field in characteristic p. These conjectures led to the development of l-adic etale cohomology as an analog of singular rational cohomology in topology by Grothendieck and his school and culminated in the proof of the Weil conjectures by Deligne in the 70s. After giving a brief introduction into the classical Weil conjectures for algebraic varieties and into moduli problems, I will outline how an analog of these Weil conjectures for the moduli stack of vector bundles on a given algebraic curve can be formulated and proved. The result basically tells "how many" vector bundles (up to isomorphisms) there are over an algebraic curve in characteristic p.

Mar 29

Tue

Constanze Roitzheim (Glasgow)

Topology Seminar

15:00

Simplicial, stable and local framings

Hicks Seminar Room J11

Abstract:
One key objective in stable homotopy theory is finding Quillen functors between model categories. These are functors respecting homotopy structures. Framings provide a way to construct and classify Quillen functors from simplicial sets to any given model category. There is also a more structured set-up where one studies Quillen functors from spectra to a stable model category. We will investigate how this is compatible with Bousfield localisations and how it can be used to study the deeper structure of the stable homotopy category.

May 3

Tue

Neil Strickland (Sheffield)

Topology Seminar

15:00

Hicks Seminar Room J11

Sep 26

Mon

Paul Mitchener (Sheffield)

Topology Seminar

15:00

Analytic K-theory vs. Algebraic K-theory

Hicks Seminar Room J11

Abstract:
In this talk we show how algebraic K-theory can be presented in a "topological" way, meaning both algebraic K-theory and analytic K-theory are obtained from the same machinery. I'm hoping that the seminar will be fairly accessible to non-specialists in K-theory.

Oct 3

Mon

David Barnes (Sheffield)

Topology Seminar

15:00

E-local Framings

Hicks Seminar Room J11

Abstract:
Framings provide a way to construct homotopically interesting functors from simplicial sets to any given model category. A more structured set-up studies stable frames, giving Quillen functors from spectra to stable model categories. We will investigate how this is compatible with Bousfield localisation to gain insight into the deeper structure of the stable homotopy category. We further show how these techniques relate to rigidity questions and how they can be used to study algebraic model categories.

Oct 10

Mon

Pokman Cheung (Sheffield)

Topology Seminar

15:00

A geometric description of the Witten genus

Hicks Seminar Room J11

Abstract:
The study of elliptic genera and elliptic cohomology, which started in the 80s, has provided interactions between such areas as homotopy theory, elliptic curves \& modular forms, topology \& geometry of free loop spaces, and mathematical structure of quantum field theory. Roughly speaking, this topic is like a higher version of topological K-theory and index theory. However, unlike its classical counterpart, this higher version still lacks a geometric interpretation, which has been a central problem of the topic. I will discuss some recent work towards this goal.

Oct 17

Mon

Arjun Malhotra (Muenster)

Topology Seminar

15:00

Spin(c) bordism of elementary abelian groups

Hicks Seminar Room J11

Abstract:
The Gromov-Lawson-Rosenberg conjecture for a group G says that a spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a topological obstruction lying in the real connective k-theory of G vanishes. I will indicate how we construct explicit spin projective bundles to prove the conjecture for elementary abelian groups, and discuss how the problem can be reduced to describing the complex connective k-theory via spin-c projective bundles.

Oct 24

Mon

Nora Seeliger (Oberwolfach)

Topology Seminar

15:00

Group models for fusion systems and cohomology

Hicks Seminar Room J11

Abstract:
Fusion systems are categories modelled on the conjugacy relations of a Sylow p-subgroup in a finite group. Every finite group gives rise to a fusion system for every prime dividing its order however there are fusion systems which cannot be realized as a fusion system of any finite group. This led to the concept of an exotic fusion system. In 2007 Robinson and independently Leary-Stancu constructed infinite groups realizing arbitrary fusion systems, a third one is due Libman-Seeliger in 2009. In this talk we will present a new model realizing arbitrary fusion systems and discuss some of its properties and moreover compare the cohomology of all these group models to the cohomology of the fusion system.

Oct 31

Mon

Nick Gurski (Sheffield)

Topology Seminar

15:00

Icons

Hicks Seminar Room J11

Abstract:
The first thing many people learn about higher categories is that monoidal categories are just 2-categories with a single object. This statement is supposed to prepare you for learning about 2-categories, since monoidal categories are extremely common. As with many not-quite-theorems in category theory, the truth of this statement depends on what the word "are" means. This talk is intended to introduce some basic concepts in the study of 2-categories (or maybe even n-categories in general), with one goal being to discuss the icons of the title and why they are interesting.

Nov 14

Mon

Sarah Whitehouse (Sheffield)

Topology Seminar

15:00

Derived A-infinity algebras from the point of view of operads

Hicks Seminar Room J11

Abstract:
A-infinity algebras arise whenever one has a multiplication which is ``associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras (dgas) via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general ground ring. I will put derived A-infinity algebras into the context of operads and show that the operad for derived A-infinity algebras can be viewed as a free resolution of the operad for bidgas, in the same sense that the A-infinity operad is a free resolution of the operad for dgas. This is joint work with Muriel Livernet and Constanze Roitzheim. \\ Cake will be provided by Jonathon

Nov 21

Mon

Danny Stevenson (Glasgow)

Topology Seminar

15:00

A classical construction for simplicial sets revisited

Hicks Seminar Room J11

Abstract:
Simplicial sets became popular in the 1950s as a combinatorial way to study the homotopy theory of topological spaces. They are more robust then the older notion of simplicial complexes, which were introduced for the same purpose. We will review some functors arising in the theory of simplicial sets, some well-known, some not-so-well-known, and show how the latter give a very useful perspective on the Kan loop group functor. We will also describe a generalized Cartier-Dold-Puppe theorem for simplicial sets, and show how this leads to a very simple proof of a classical theorem of Kan. \\ Cake will be provided by Vikki

Nov 28

Mon

Jacob Rasmussen (Cambridge)

Topology Seminar

15:00

Torus knots, Hilbert schemes, and Khovanov homology

Hicks Seminar Room J11

Abstract:
Khovanov homology is an invariant of knots in S^3 which generalizes the Jones polynomial. I'll discuss some conjectures which relate the Khovanov homology of torus knots to some objects in algebraic geometry (Hilbert schemes of singular curves) and algebra (rational Cherednik algebras). Joint work with E. Gorsky, A. Oblomkov, and V. Shende. \\ Cake will be provided by Matt

Dec 12

Mon

Roald Koudenburg (Sheffield)

Topology Seminar

15:00

Homotopy theory for generalised algebraic operads and their algebras

Hicks Seminar Room J11

Abstract:
The homotopy theory for classical operads and algebras over them is well understood. In more detail: we know what homotopy algebras are, how they can be transferred along weak equivalences and when they can be rectified to strict algebra structures. To start with we will recall these notions and results, working throughout in the category of chain complexes over a field of charateristic zero.

We will then define classical operads as symmetric monoidal functors, as introduced by E. Getzler. Using this approach we can easily generalise to structures in which operations have multiple outputs (properads) or where the distinction between inputs and outputs is removed (cyclic operads). Following this we will think about how to obtain model structures on categories of such generalised operads, as well as on the categories of their (lax) algebras.

Cake will be provided by Eugenia

Feb 9

Thu

David Barnes (Sheffield)

Topology Seminar

15:00

Stable Model Categories

Hicks Seminar Room J11

Abstract:
A model category is a way of giving a category a notion of homotopy. Hence in a model category we can talk of maps being homotopic or objects being homotopy equivalent. The two basic examples of model categories are topological spaces and chain complexes. Hence model categories are of interest to both topologists and algebraists. One condition that a model category may satisfy is that of stability. This is where there is a shift functor or suspension functor which is an equivalence on the homotopy category. Chain complexes are such an example, however the category of topological is not a stable model category. In this talk I will define the notion of stability more carefully, and try to describe how one may alter a category to make it stable. In particular, we will see that spectra are the stabilisation of spaces.

Feb 16

Thu

Fionntan Roukema (Sheffield)

Topology Seminar

15:00

Dehn Fillings of Manifolds with Small Volume 2

Hicks Seminar Room J11

Abstract:
In this talk we will recall some basic notions from Dehn surgery and remind ourselves about why we care about ``exceptional surgeries'' and ``exceptional pairs''. We then return to a tabulation of 3-manifolds of ``small volume'' and speak how it is possible to enumerate the set of exceptional slopes, pairs and fillings of ``most'' manifolds in this tabulation. If time permits we will speak about questions for future consideration. \\ Cake will be provided by Eugenia

Feb 23

Thu

Eugenia Cheng (Sheffield)

Topology Seminar

15:00

Multivariable adjunctions and mates

Hicks Seminar Room J11

Abstract:
(Joint work with Nick Gurski and Emily Riehl.) The so-called ``mates correspondence'' (named by Australians) arises in the presence of adjunctions. It enables us neatly to pass between natural transformations involving left adjoints and those involving right adjoints, and is used efficaciously in Emily Riehl's work on algebraic model categories. When Emily visited us last year, she was extending her work to algebraic monoidal model categories. For this, she was looking for a multivariable generalisation of the mates correspondence, and a framework in which to describe it. The ordinary mates correspondence is elegantly described using double categories, and Nick and I sat down with Emily and produced the theory of ``cyclic double multicategories'', which not only answers her question but is also a satisfying piece of category theory: the best of both worlds. Moreover, it is an output directly resulting from MSRC funding.

Mar 1

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

15:00

On categories with two objects

Hicks Seminar Room J11

Abstract:
In this talk we'll analyse cofibrant objects in the model category of categories on two objects enriched in a monoidal model category. As an application, we will obtain a Bergner type model structure on the category of all such enriched categories with arbitrary set of objects.

Cake will be provided by Jonathon

Mar 8

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

15:00

Two models for infinity-operads

Hicks Seminar Room J11

Abstract:
I will explain the Lurie model category for infinity operads based on the theory of marked simplicial sets over the nerve of Gamma, the model category for infinity operads based on dendroidal sets which I introduced with Cisinski, and a comparison between the two. \\ Cake will be provided by Matt

Mar 15

Thu

Simona Paoli (Leicester)

Topology Seminar

15:00

Hicks Seminar Room J11

Abstract:
Cake will be provided by Vikki

Mar 22

Thu

Philipp Wruck (Sheffield)

Topology Seminar

15:00

Equivariant Transversality: Overview and Recent Developments

Hicks Seminar Room J11

Abstract:
The notion of transversality allows us to successfully describe generic behaviour of smooth maps and has important impacts in various branches of topology. A simple adaption in the equivariant context is not possible, but using techniques from real algebraic geometry and the theory of stratified spaces, a natural concept of equivariant transversality has been developed. We sketch the basic ideas and give some applications of equivariant transversality. Then we show how these ideas can be adapted to define a notion of equivariant non-degeneracy, which is important for the investigation of fixed orbits of equivariant maps and their relation to equivariant homotopy invariants. \\ Cake will be provided by Thomas

Mar 29

Thu

Ian Leary (Southampton)

Topology Seminar

15:00

Platonic polygonal complexes II

Hicks Seminar Room J11

Abstract:
A flag in a polygonal complex is a triple consisting of a mutually incident vertex, edge and polygon. A polygonal complex is said to be platonic if it admits a flag transitive group of symmetries. In this talk I shall go into more detail concerning the classification of some families of platonic polygonal complexes, focusing especially on the (rather degenerate) cases when the polygons have 3, 4 or 5 sides. (The original parts of this talk are joint work with T Januszkiewicz, R Valle and R Vogeler.)

Cake will be provided by Sarah

Apr 26

Thu

Martin Crossley (Swansea)

Topology Seminar

15:00

Conjugation Invariants in the Adem-free Steenrod algebra

Hicks Seminar Room J11

Abstract:
In work with Sarah Whitehouse we attempted to calculate the invariants of the mod 2 dual Steenrod algebra under the Hopf algebra conjugation. In work with Deniz Turgay we now tackle this problem by removing the Adem relations and working with a free associative algebra instead. We give a description of the linear structure of the conjugation invariants there, and comment on the remaining problem of deriving information on the Steenrod algebra.

May 3

Thu

Andrew Lobb (Durham)

Topology Seminar

15:00

Two-strand twisting and knot homologies

Hicks Seminar Room J11

Abstract:
We give an introduction to some quantum knot homologies and show how twisting up a pair of adjacent strands in a knot, combined with some straightforward homological algebra, allows us to deduce some interesting consequences.

May 17

Thu

Ivan Panin (St.Petersburg Department of V.A.Steklov Institute of Mathematics of the Russian Academy of Sciences)

Topology Seminar

15:00

Construction of the triangulated category DK_(k) of K-motives

Hicks Seminar Room J11

Abstract:
We construct a triangulated category $DK_{(k)}$ of $K$-motives in the style of Voevodsky's construction of the category $DM(k)$. Each smooth $k$-variety has its $K$-motive $M_K(X)$ in the category $DK_{(k)}$ of $K$-motives and $\text{Hom}(M_K(X),M_K(pt)[n])=K_n(X)$, where $pt=Spec(k)$ and $K_n(X)$ is Quillen's $K$-groups of $X$. The $K$-motive $M_K(pt)$ of the point has a natural Grayson's "filtration". Due to Suslin's results successive cones of the "filtration" are the motivic complexes $Z(n)$. This observation gives rise to a new construction of a spectral sequence which starts at motivic cohomology of a smooth variety $X$ and converges to its Quillen $K$-groups. The results have been obtained joint with G. Garkusha.

Sep 24

Mon

Paul Mitchener (Sheffield)

Topology Seminar

15:00

Semigroup Algebras and Homology

Hicks Seminar Room J11

Abstract:
In this talk, we look at the question of how we might compute the K-theory of a semigroup $C^*$-algebra. On the way, we look at a few features of equivariant homology for semigroups. I intend to take an elementary approach here, introducing all relevant concepts.

Oct 1

Mon

David Barnes (Sheffield)

Topology Seminar

15:00

Localisations of Stable Model Categories

Hicks Seminar Room J11

Abstract:
Most of the homotopy theories that we are interested in are extremely complicated and it is hard to discern patterns in this data. To remedy this, we often discard some of the information of the homotopy theory in return for more structure. The canonical way of doing so is Bousfield localisation. In this talk I will introduce the notion of Bousfield localisations of model categories and show how in the stable case these localisations are very simple to construct.

Oct 8

Mon

Constanze Roitzheim (Kent)

Topology Seminar

15:00

Modular Rigidity of E-local Spectra

Hicks Seminar Room J11

Abstract:
One key objective in stable homotopy theory is finding Quillen functors between model categories. Stable frames provide a way to construct and classify Quillen functors from spectra to any given stable model category. Furthermore, they equip the homotopy category of a stable model category with a module structure over the stable homotopy category Ho(Sp). We will investigate how this is compatible with Bousfield localisations and how it can be used to study the deeper structure of the stable homotopy category. We will then see that the Ho(Sp)-module structure completely determines the homotopy type of the E-local stable homotopy category for any homology theory E.

Cake will be provided by Vikki

Oct 15

Mon

Pokman Cheung (Sheffield)

Topology Seminar

15:00

Spinors on formal loops

Hicks Seminar Room J11

Abstract:
tba

Oct 29

Mon

Ines Henriques (Sheffield)

Topology Seminar

15:00

Quasi-complete intersections

Hicks Seminar Room J11

Abstract:
Over a local ring $R$, we define an ideal $I$ to be quasi-complete intersection if the homology of the Koszul complex $E$ on a generating set of $I$ is free as a module over $S = R/I$, and the canonical map of graded S-algebras $\bigwedge_{*}^{S} ( H_{1} (E))$ â†’ $H_{*} (E)$ is bijective.

This class of ideals strictly contains the class of complete intersection (c.i.) ideals. The simplest type of quasi-c.i. ideals that are not complete intersections are generated by one exact zero-divisor.

We will discuss the behavior of some basic homological and structural invariants with respect to the change of rings $R \to S$. Several basic invariants of $R$ determine those of the residue ring $R/I$ and recover the formulas that hold in the particular case when $I$ is generated by a regular sequence. Under additional hypothesis, we conclude that $R$ and $S$ are equally far from being Cohen-Macaulay, Gorenstein, or complete intersection.

Nov 5

Mon

Marcy Robertson (Western Ontario)

Topology Seminar

15:00

On Topological Triangulated Orbit Categories

Hicks Seminar Room J11

Abstract:
In 2005, Keller showed that the orbit category associated to the bounded derived category of a hereditary category under an auto equivalence is triangulated. As an application he proved that the cluster category is triangulated. We show that this theorem generalizes to triangulated categories with topological origin (i.e. the homotopy category of a stable model category). As an application we construct a topological triangulated category which models the cluster category. This is joint work with Andrew Salch.

Nov 19

Mon

Jon Woolf (Liverpool)

Topology Seminar

15:00

Whitney Categories and the Tangle Hypothesis

Hicks Seminar Room J11

Abstract:
Baez and Dolan's Tangle Hypothesis is that 'higher categories of tangles' have an algebraic characterisation as 'free multiply-monoidal categories with duals'. I will try to explain what this means and to make it precise within the context of `Whitney categories'. These are a geometric notion of 'higher category with duals', based on Whitney stratified spaces. I will then sketch how the Tangle Hypothesis for Whitney categories reduces to the Pontrjagin-Thom construction. This is joint work with Conor Smyth.

Nov 20

Tue

Irakli Patchkoria (Bonn)

Topology Seminar

17:00

Rigidity in equivariant stable homotopy theory

Hicks Seminar Room J11

Abstract:
Let G be a ﬁnite abelian group or ﬁnite (non-abelian) 2-group. We show that the 2-local G-equivariant stable homotopy category, indexed on a complete G-universe, has a unique G- equivariant model in the sense of Quillen model categories. This means that the suspension functor, homotopy coﬁber sequences and the stable Burnside category determine all "higher order structure" of the 2-local G-equivariant stable homotopy category such as for example equivariant homotopy types of function G-spaces. The theorem can be seen as an equivariant generalization of Schwede's rigidity theorem at prime 2.

Nov 26

Mon

Andrew Stacey (Trondheim)

Topology Seminar

15:00

That which we call a manifold ...

Hicks Seminar Room J11

Abstract:
It's well known that the mapping space of two finite dimensional manifolds can be given the structure of an infinite dimensional manifold modelled on Frechet spaces (provided the source is compact). However, it is not that the charts on the original manifolds give the charts on the mapping space: it is a little bit more complicated than that. These complications become important when one extends this construction, either to spaces more general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to transport local properties of a space to local properties of its mapping space. As an application, we shall show that applying the mapping construction to a regular map is again regular.

Note: the theme of this talk is the same as a talk I gave in Sheffield a little over a year ago so this can be thought of as a report on how my ideas have developed over the intervening time. I shan't assume that anyone remembers the original talk, whilst for anyone who does then there is definite progress to report.

Cake will be provided by Philipp

Dec 3

Mon

Dmitry Kaledin (Steklov Institute of Mathematics)

Topology Seminar

15:00

Derived Mackey functors

Hicks Seminar Room J11

Abstract:
Mackey functors associated to a finite group $G$ appear both in equivariant stable homotopy theory and in finite group theory, and are quite useful in both areas. Since Mackey functors form an abelian category, one can consider its derived category. However, I going to argue that there is a better alternative: a triangulated category containing the abelian category of Mackey functors but different from its derived category, with a better behavior, more natural definition, and more closely approximating equivariant stable homotopy category. Moreover, our derived Mackey exist in bigger generality, and what is the natural counterpart of this in stable homotopy seems to be an interesting question.

Cake will be provided by Pokman

Dec 10

Mon

Tom Leinster (Edinburgh)

Topology Seminar

15:00

Entropy is inevitable

Hicks Seminar Room J11

Abstract:
The title refers not to the death of the universe, but to the fact that the concept of entropy is present in the pure-mathematical heartlands of algebra and topology, whether we like it or not. I will describe a categorical machine which, when fed as input the concepts of topological simplex and real number, produces as output the concept of Shannon entropy. The most important component of this machine is the notion of "internal algebra" in an algebra for an operad (generalizing the notion of monoid in a monoidal category). The resulting characterization of Shannon entropy can be stripped completely of its categorical garb, to obtain a simple, new, and entirely elementary characterization. This last theorem is joint work with John Baez and Tobias Fritz.

Cake will be provided by Callan

Feb 4

Mon

John McCleary (Vassar College)

Topology Seminar

15:00

Topology for Combinatorics

Hicks Seminar Room J11

Abstract:
Topology studies spaces that include spaces of all possible configurations of combinatorial problems. Often the configurations come with symmetry and the problem at hand can be rewritten as a linear condition on a test map. Within this framework, topological methods can be made to give concrete combinatorial results. In joint work with Pavle and Alexandra Blagojevic, we use algebraic topology to carry out this method.

Feb 14

Thu

Muriel Livernet (UniversitÃ© Paris 13)

Topology Seminar

15:00

On the homology of the Swiss-Cheese operad

Hicks Seminar Room J11

Abstract:
In this talk I will define the Swiss-cheese operad (a combination of the little discs and the interval operad), and show our main resuls:

the homology of the Swiss-cheese operad is Koszul
the spectral sequence associated to the geometric Swiss-cheese operad degenerates at $E_2$.

In order to do this I will briefly explain how it has been done for the little discs operad, and point out the difficulties inherent to the Swiss-cheese operad.

Feb 21

Thu

Christine Vespa (University of Strasbourg)

Topology Seminar

15:00

Stable homology of groups with polynomial coefficients

Hicks Seminar Room J11

Abstract:
We say that the homology of a sequence of groups $(G_n)$ stabilizes if the homology groups, of each degree, of the groups $G_n$ is independent of n, for n big enough. Stability with constant coefficients or more generally polynomial coefficients has been proved for many families of groups. In this talk I will consider the question of the computation of this stable value. In particular, I will present the following recent result obtained in collaboration with AurÃ©lien Djament: the stable homology of automorphism groups of free groups with coefficients given by a polynomial covariant functor like the abelianization or any tensor power of it, is trivial.

Feb 28

Thu

Nadia Gheith (Sheffield)

Topology Seminar

15:00

Coarse Cofibration Category

Hicks Seminar Room J11

Abstract:
Baues introduced a notion of cofibration category as a generalisation of a Quillen model category. He defined it to be a category together with two classes of morphisms called cofibrations and weak equivalences such that specific axioms are satisfied. In this talk I will introduce a notion of closeness equivalence classes of coarse maps-these are maps between spaces preserving the large scale structure. And prove that the category of spaces and closeness equivalence classes with two classes of morphisms called coarse cofibration classes and coarse homotopy equivalence classes satisfy the cofibration category axioms. This category will be called the Coarse cofibration category.

Mar 7

Thu

Ralph Kaufmann (Purdue University)

Topology Seminar

15:00

Feynman categories

Hicks Seminar Room J11

Abstract:
There is a plethora of operad type structures and constructions which arise naturally in classical and quantum contexts such as operations on cochains, string topology or Gromov-Witten invariants. We give a novel categorical framework which allows us to handle all these different beasts in one simple fashion. In this context, many of the relevant constructions are simply Kan extensions. We are also able to show how in this framework bar constructions, Feynman transforms, master and BV equations appear naturally.

Mar 14

Thu

Reiner Lauterbach (University of Hamburg)

Topology Seminar

15:00

Equivariant Bifurcation and Ize Conjecture

Hicks Seminar Room J11

Apr 11

Thu

John Greenlees (Sheffield)

Topology Seminar

15:00

THH and the Gorenstein condition

Hicks Seminar Room J11

Abstract:
Calculations of Boekstedt and Ausoni give examples showing that with suitable coefficients $THH(R)$ has strong duality properties. The talk will describe how to establish these duality properties without a complete calculation, showing that THH of ring spectra has Gorenstein duality remarkably often. The context is the notion of Gorenstein ring spectra studied with Dwyer and Iyengar, and the talk will include a suitable summary.

Apr 18

Thu

Simon Covez (University of Luxembourg)

Topology Seminar

15:00

On the conjectural Leibniz homology for groups

Hicks Seminar Room J11

Abstract:
Twenty years ago Jean-Louis Loday has introduced and studied Leibniz algebras and their homology theory. Following this discovery, he has conjectured the existence of a conjectural Leibniz homology for groups and some of its properties, such as the existence of an algebraic structure on this conjectural homology or the existence of a natural morphism from this conjectural homology to the usual homology theory of groups. In this talk we will see that the homology theory of racks satisfies most of these properties and, therefore, should be this conjectural Leibniz homology theory.

May 2

Thu

Oscar Randal-Williams (Cambridge)

Topology Seminar

15:00

Infinite loop spaces and positive scalar curvature

Hicks Seminar Room J11

Abstract:
It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research. I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and the speaker to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$ of dimension at least 6 is as complicated as can possibly be detected by index-theory. This is joint work with Boris Botvinnik and Johannes Ebert.

May 9

Thu

Simon Willerton (Sheffield)

Topology Seminar

15:00

Integral transforms, correspondences and profunctors

Hicks Seminar Room J11

May 16

Thu

Mark Grant (Nottingham)

Topology Seminar

15:00

Topological complexity of braid groups

Hicks Seminar Room J11

Abstract:
Topological complexity (TC) is a numerical homotopy invariant which quantifies the complexity of navigation in a topological space. Defined by Michael Farber in the early 21st century, it gives topological information about the motion planning problem in robotics. Briefly, TC(X) is the sectional category of the free path fibration on X. An interesting open problem is to determine TC of a K(G,1)-space algebraically in terms of the fundamental group G. After surveying this problem and related results, we will present an approach to finding lower bounds which is purely algebraic. We will then discuss how this can be applied to estimate the topological complexity of braid groups. This is joint work with Greg Lupton and John Oprea.

Oct 3

Thu

Paul Mitchener (Sheffield)

Topology Seminar

16:00

Discrete homotopy and homology

Hicks Seminar Room J11

Abstract:
In this article we introduce discrete analogues of homotopy and homology groups on a particular scale, and state and maybe prove some analogues of some of the classic theorems of algebraic topology. We also make an attempt to compare what happens in the limit as the scale gets larger and larger with some of the corresponding groups in coarse geometry.

Oct 10

Thu

Fionntan Roukema (Sheffield)

Topology Seminar

16:00

Enumerating Exceptional Knot Complement Pairs

Hicks Seminar Room J11

Abstract:
Enumerating exceptional pairs (cusped hyperbolic manifolds with distinct non-hyperbolic fillings) is a natural and well studied programme in the literature. In this talk we will restrict our attention to hyperbolic knot complements in S^3. We will see that this essentially reduces to the study of Berge knots, and we will think about an approach to performing a complete enumeration of exceptional pairs in this setting.

Oct 17

Thu

Philipp Wruck (Sheffield)

Topology Seminar

16:00

Using tom Dieck functors to obtain global Tambara functors

Hicks Seminar Room J11

Abstract:
Many equivariant homology theories are definable not just for a particular group but for every compact Lie group. Such theories can be represented by global spectra. For a fixed group $G$, an ordinary equivariant homology theory is essentially the same as a $G$-Mackey functor, and $\pi_0$ of a $G$-spectrum naturally carries the structure of a $G$-Mackey functor. Therefore it is resonable to ask for a global equivalent of Mackey functors with similar properties. An important question is in what way additional structure in the spectrum translates into properties of the Mackey functor, e.g. when the spectrum is a commutative ring spectrum. The resulting structure in this case is called a Tambara functor. For finite groups, this structure is well understood. For compact Lie groups, Schwede has recently provided some insight with his notion of global power functors. In this talk, we will give an overview of the basic ideas of the theory of Mackey functors, Tambara functors and their global equivalents. We will present an approach based on work of tom Dieck which circumvents the use of stable homotopy theory to define global functors for compact Lie groups. This recovers the results of Schwede when passing to a suitable quotient.

Oct 24

Thu

Pokman Cheung (Sheffield)

Topology Seminar

16:00

Factorisation algebras and factorisation homology

Hicks Seminar Room J11

Abstract:
This talk will be an overview of the theory of factorisation algebras. Factorisation algebras provide a local-to-global machinery (like, but also unlike, sheaves) and arise in the study of e.g. homotopy commutative algebras, mapping spaces and quantum field theory. I will discuss some examples in topology, geometry and (perhaps) mathematical physics.

Nov 7

Thu

Alexander Vishik (Nottingham)

Topology Seminar

16:00

Symmetric and Steenrod operations in algebraic cobordism

Hicks Seminar Room J11

Abstract:
Symmetric operations are encoding all integral divisibilities of characteristic numbers of algebraic varieties. This permits to apply them to various questions related to torsion effects, getting more subtle results than what Landweber-Novikov operations would give. They also define natural obstructions for presenting a cobordism element by the class of an embedding. These operations are closely related to Steenrod operations in Algebraic Cobordism. There are two types of those: type of Quillen, and type of Tom Dieck. The latter are substantially more subtle, and were constructed only recently.

Feb 13

Thu

Andrey Lazarev (Lancaster)

Topology Seminar

15:00

Derived localization of algebras and modules

Hicks Seminar Room J11

Abstract:
The concept of localization permeates homotopy theory: homotopy categories are constructed by localizing closed model categories at weak equivalences. Localization of commutative rings and their modules is an exact functor and is well-understood. In contrast, localization of noncommutative algebras is a more subtle procedure since it needs to be derived to have good properties. In this talk I discuss the notion of derived localization of algebras and prove that it can be constructed as an appropriate Bousfield localization in the category of modules. As an application I obtain a very general version of the group completion theorem and a derived Riemann-Hilbert correspondence. This is joint work with Joe Chuang and Chris Braun.

Feb 20

Thu

Thomas Cottrell (Sheffield)

Topology Seminar

15:00

Weak n-categories: algebraic versus non-algebraic definitions

Hicks Seminar Room J11

Abstract:
An n-category is a type of higher-dimensional category which, as well as having objects and morphisms, has 2-morphisms between the morphisms, 3-morphisms between the 2-morphisms, and so on, up to n-morphisms for some fixed natural number n. In a strict n-category, composition of these morphisms is associative and unital. The strict case is well-understood, but strict n-categories are not suitable for describing situations in which composition is not associative and unital, such as concatenation of paths and homotopies. For this a notion of weak n-category is required, in which composition is only associative and unital up to some higher-dimensional cells. Weak n-categories share a close relationship with topology via the homotopy hypothesis of Grothendieck. Many definitions of weak n-category have been proposed, but the relationships between these definitions are not yet well understood. These definitions can be divided into two types: algebraic definitions and non-algebraic definitions. In this talk I will explain what these terms mean, and give two examples of definitions. The first of these is an algebraic definition, due to Penon, which uses the theory of monads; the second is a non-algebraic definition, due to Tamsamani and Simpson, which takes a simplicial approach. I will finish by describing some of my work towards comparing these two definitions.

Mar 6

Thu

Moritz Groth (Nijmegen)

Topology Seminar

15:00

Grothendieck derivators (and tilting theory)

Hicks Seminar Room J11

Abstract:
The theory of derivators (going back to Grothendieck, Heller, and others) provides an axiomatic approach to homotopy theory. It adresses the problem that the rather crude passage from model categories to homotopy categories results in a serious loss of information. In the stable context, the typical defects of triangulated categories (non-functoriality of cone construction, lack of homotopy colimits) can be seen as a reminiscent of this fact. The simple but surprisingly powerful idea behind a derivator is that instead one should form homotopy categories of various diagram categories and also keep track of the calculus of homotopy Kan extensions. In this talk I will give an introduction to derivators, indicating that stable derivators provide an enhancement of triangulated categories. If time permits, I will sketch some applications to tilting theory.

Mar 13

Thu

Ieke Moerdijk (Sheffield/Nijmegen)

Topology Seminar

15:00

The homotopy colimit functor as a Quillen equivalence

Hicks Seminar Room J11

Abstract:
Let $A$ be a small category. I will present an elementary proof of the fact that the homotopy colimit functor from $A$-diagrams of spaces to spaces over the nerve of $A$ provides a left Quillen equivalence between appropriate model category structures (joint work with Gijs Heuts).

Mar 27

Thu

Neil Strickland (Sheffield)

Topology Seminar

15:00

A large diagram in unstable homotopy theory

Hicks Seminar Room J11

Abstract:
I will discuss a diagram involving an odd-primary analogue of the EHP sequence, the double suspension map and so on. Almost all ingredients appear in various places in the literature, but they are not combined into a single diagram. Moreover, many of the spaces and maps are constructed in a way that involves extensive choices. There are many issues about compatibility of choices that do not seem to be very clear.

This talk will aim to explain some background and to describe a family of interesting problems; there will not be many actual results.

Apr 3

Thu

Tobias Dyckerhoff (Oxford)

Topology Seminar

15:00

Triangulated surfaces in triangulated categories

Hicks Seminar Room J11

Abstract:
Given a triangulated category $A$, equipped with a differential $\mathbb{Z}/2$-graded enhancement, and a triangulated oriented marked surface $S$, we explain how to define a space $X(S,A)$ which classifies systems of exact triangles in $A$ parametrized by the triangles of $S$. The space $X(S,A)$ is independent, up to essentially unique homotopy equivalence, of the choice of triangulation and is therefore acted upon by the mapping class group of the surface. We can describe the space $X(S,A)$ as a mapping space $Map(F(S),A)$, where $F(S)$ is the universal differential $\mathbb{Z}/2$-graded category of exact triangles parametrized by $S$. It turns out that $F(S)$ is a purely topological variant of the Fukaya category of $S$. Our construction of $F(S)$ can then be regarded as implementing a 2-dimensional instance of Kontsevich's proposal on localizing the Fukaya category along a singular Lagrangian spine. As we will see, these results arise as applications of a general theory of cyclic 2-Segal spaces. This talk is based on joint work with Mikhail Kapranov.

Apr 30

Wed

Jeffrey Giansiracusa (Swansea)

Topology Seminar

16:00

$G$-equivariant open-closed TCFTs

Hicks Seminar Room J11

Abstract:
Open 2d TCFTs correspond to cyclic $A_\infty$ algebras, and Costello showed that any open theory has a universal extension to an open-closed theory in which the closed state space (the value of the functor on a circle) is the Hochschild homology of the open algebra. We will give a $G$-equivariant generalization of this theorem, meaning that the surfaces are now equipped with principal $G$-bundles. Equivariant Hochschild homology and a new ribbon graph decomposition of the moduli space of surfaces with $G$-bundles are the principal ingredients. This is joint work with Ramses Fernandez-Valencia.

May 15

Thu

Frank Neumann (Leicester)

Topology Seminar

15:00

Étale homotopy theory of algebraic stacks

Hicks Seminar Room J11

Abstract:
I will give an overview on étale homotopy theory à la Artin-Mazur of Deligne-Mumford stacks and discuss several examples including moduli stacks of algebraic curves and principally polarised abelian varieties and their compactifications. If time permits I will indicate how to extend the machinery to Artin stacks and how to apply it to the moduli stack of principal bundles over a curve.

May 20

Tue

Rosona Eldred (Muenster)

Topology Seminar

15:00

Goodwillie calculus and nilpotence

Hicks Seminar Room J11

Abstract:
The Goodwillie Taylor tower of a functor is a filtration with layers built from spectra. In particular, linear functors look roughly like (spectrum) $\wedge$ (input). Thinking of spectra as the abelianization of topological spaces, we can then ask how close this tower is to being a sort of nilpotent filtration for a functor, like the lower central series filtration of a group. I will give some background on the relationship between nilpotence and the Goodwillie tower and talk about my work tying in the partial-towers, formulated in terms of a decomposition involving adjoint functors.

Oct 16

Thu

John Greenlees (Sheffield)

Topology Seminar

16:00

The localization theorem and algebraic models of rational equivariant cohomology theories

Hicks Seminar Room J11

Oct 23

Thu

Magda Kedziorek (Sheffield)

Topology Seminar

16:00

tbc

Hicks Seminar Room J11

Oct 30

Thu

Neil Strickland (Sheffield)

Topology Seminar

16:00

An introduction to Homotopical Type Theory

Hicks Seminar Room J11

Abstract:
I will give an introduction to Voevodsky's Homotopical Type Theory (HTT), and attempt to reconcile the following perspectives:

HTT provides an intrinsic language for talking about homotopical phenomena, independent of any underlying geometric category or model category.
HTT is a natural framework for thinking about computer representation of mathematical objects, propositions and proofs, where equality must usually be checked by a nontrivial computation.

Nov 6

Thu

Dimitar Kodjabachev (Sheffield)

Topology Seminar

16:00

A strictly commutative model for E-infinity quasi-categories

Hicks Seminar Room J11

Abstract:
I will show that E-infinity quasi-categories can be rigidified to strictly commutative objects in the larger category of diagrams of simplicial sets indexed by finite sets and injections. This complements earlier work on diagram spaces by Steffen Sagave and Christian Schlichtkrull.

Nov 20

Thu

Paul Mitchener (Sheffield)

Topology Seminar

16:00

A menagerie of assembly maps

Hicks Seminar Room J11

Abstract:
In geometric topology, a number of different maps are referred to as assembly maps, and various conjectures are present which assert that instances of these maps are injective. A theorem due to Weiss and Williams in the 1990s describes and characterises assembly in terms of spectra. In this talk, we look at a refinement of this machinery which lets us not just characterise assembly maps but give "universal" proofs of injectivity which apply to a number of different situations. We will conclude by talking about examples of the machinery, possibly including C*-algebra K-theory, L-theory, algebraic K-theory, and homotopy algebraic K-theory.

Nov 27

Thu

Vesna Stojanoska (MPI Bonn)

Topology Seminar

16:00

Arithmetic duality for generalized cohomology theories

Hicks Seminar Room J11

Abstract:
Poitou-Tate duality is a duality for the Galois cohomology of finite modules over the absolute Galois group of a global field. This arithmetic duality is reminiscent of Poincaré duality for manifolds familiar to topologists. In joint work with Tomer Schlank we upgrade it to a duality for generalized cohomology theories with action by such an absolute Galois group. We believe this upgraded duality should lead to a better understanding of rational points on algebraic varieties.

Dec 4

Thu

Piotr Pstragowski (Sheffield)

Topology Seminar

16:00

On the Cobordism Hypothesis and the grammar of space

Hicks Seminar Room J11

Abstract:
During the talk, I will explain the concept of an extended topological field theory and formulate the Cobordism Hypothesis, now a theorem of Jacob Lurie. I will give some examples of "grammar of space" phenomena, where a geometrically defined structure turns out to be universal in some strong algebraic sense.

Feb 16

Mon

Sarah Whitehouse (Sheffield)

Topology Seminar

16:00

A-infinity algebras and spectral sequences

Hicks Seminar Room J11

Abstract:
This will be an expository talk, on the connection between A-infinity algebras and multiplicative spectral sequences. Since the cohomology of a dga over a field has an A-infinity algebra structure, there must be some kind of A-infinity structure on the pages of a multiplicative spectral sequence. I will review some work of Lapin and Herscovich in this direction.

Feb 23

Mon

David O'Sullivan (Sheffield)

Topology Seminar

16:00

Bundles in Noncommutative Topology

Hicks Seminar Room J11

Abstract:
In ordinary topology we are often interested in families of objects parametrized over some base space. Vector bundles and fibrations are the obvious examples. The same is true of bundles in noncommutative topology, where they play a central role in the representation theory of topological groupoids.

In some senses our bundles are a lot more general, in that we are not bound by things like local triviality. We can therefore construct some very interesting and powerful bundle-like constructions. Perhaps the most general is the Fell bundle, which can be though of as a bundle of Banach spaces in which the base object is no longer a topological space but instead a topological groupoid.

In this talk I will explain how Fell bundles are constructed and how they are used in representation theory. It turns out that this is best done using the language of C*-categories, but with a new internal construction in the category of topological spaces. Along the way I will give an overview of the established theory of Banach- and Hilbert- bundles. I will also say a little on how we can study Fell bundles using an existing topological invariant.

Mar 9

Mon

Tom Sutton (Sheffield)

Topology Seminar

16:00

Hicks Seminar Room J11

Mar 16

Mon

Julie Bergner (UC Riverside)

Topology Seminar

16:00

Models for equivariant (\infty, 1)-categories

Hicks Seminar Room J11

Abstract:
Recent results of Marc Stephan give conditions under which a cofibrantly generated model category has an equivariant analogue, where the objects have a group action and weak equivalences and fibrations are defined via fixed point objects. We apply his results to several models for (\infty, 1)-categories. For discrete groups, all models satisfy the required conditions. For simplicial or topological groups, we need to consider those models which have the additional structure of a simplicial or topological model category, respectively. We can also give an explicit description for equivariant complete Segal spaces, leading to examples from G-categories.

Apr 13

Mon

Andrew Tonks (Leicester)

Topology Seminar

16:00

A homotopical perturbation lemma

Hicks Seminar Room J11

Abstract:
A cute 1961 paper of C.T.C. Wall shows that from free chain resolutions for groups N and Q one may construct, by a ‘twisted tensor product’, a resolution for any group extension of N by Q. More recently Brown and others have attempted, with some degree of success, to lift this construction from the category of chain complexes to that of crossed complexes, or of CW complexes. This non-abelian situation is considerably harder; one knows, for example, that there is no homological perturbation theory for crossed complexes. In this talk we will give an overview of the problem and present some new results obtained in collaboration with O.J. Gill and G. Ellis.

Apr 27

Mon

Daniel Schappi (Sheffield)

Topology Seminar

16:00

Tannaka duality and Adams Hopf algebroids

Hicks Seminar Room J11

Abstract:
Classical Tannaka duality is a duality between groups and their categories of representations. It answers two basic questions: can we recover the group from its category of representations, and can we characterize categories of representations abstractly? These are often called the reconstruction problem and the recognition problem. In the context of affine group schemes over a field, the recognition problem was solved by Saavedra and Deligne using the notion of a (neutral) Tannakian category.
This can be generalized to the context of Adams Hopf algebroids and their categories of comodules. Using the language of stacks, this generalization gives a duality between Adams stacks and their categories of quasi-coherent sheaves. I will start with an overview of classical Tannaka duality and its generalization, and I will conclude my talk with an outline how this duality can be used to interpret various geometric constructions involving Adams stacks in terms of their associated categories.

May 11

Mon

Simona Paoli (Leicester)

Topology Seminar

16:00

Weak globularity in homotopy theory and higher category theory.

Hicks Seminar Room J11

Abstract:
Spaces and homotopy theories are fundamental objects of study of algebraic topology. One way to study these objects is to break them into smaller components with the Postnikov decomposition. To describe such decomposition purely algebraically we need higher categorical structures. We describe one approach to modelling these structures based on a new paradigm to build weak higher categories, which is the notion of weak globularity. We describe some of their connections to both homotopy theory and higher category theory.

Feb 4

Thu

Jessica Banks (Hull)

Topology Seminar

16:00

Hicks Seminar Room J11

Feb 25

Thu

Simon Willerton (Sheffield)

Topology Seminar

16:00

The magnitude of odd balls

Hicks Seminar Room J11

Apr 14

Thu

Eugenie Hunsicker (Loughborough)

Topology Seminar

16:00

From Pure Maths to Data Science: How topology, geometry and analysis can help solve data challenges.

Hicks Seminar Room J11

May 5

Thu

Brendan Owens (Glasgow)

Topology Seminar

16:00

Embeddings of rational homology 4-balls

Hicks Seminar Room J11

Abstract:
Certain 3-dimensional lens spaces are known to smoothly bound 4-manifolds with the rational homology of a ball. These can sometimes be useful in cut-and-paste constructions of interesting (exotic) smooth 4-manifolds. To this end it is interesting to identify 4-manifolds which contain these rational balls. Khodorovskiy used Kirby calculus to exhibit embeddings of rational balls in certain linear plumbed 4-manifolds, and recently Park-Park-Shin used methods from the minimal model program in 3-dimensional algebraic geometry to generalise Khodorovskiy's result. The goal of this talk is to give an accessible introduction to the objects mentioned above and also to describe a much easier topological proof of Park-Park-Shin's theorem.

May 12

Thu

Nick Gurski (Sheffield)

Topology Seminar

16:00

Picard 2-categories and models for the truncated sphere spectrum

Hicks Seminar Room J11

Abstract:
A Picard n-category is a symmetric monoidal n-category in which all cells, including objects, are invertible. The Stable Homotopy Hypothesis states that Picard n-categories should be a model for the homotopy theory of stable n-types. This is known for n=0,1, and in this talk I will discuss some of the challenges moving to the n=2 case.

Oct 4

Tue

John Greenlees (Sheffield)

Topology Seminar

16:00

Rational equivariant cohomology theories and the spectrum of the sphere.

Hicks Seminar Room J11

Abstract:
Rational G-equivariant cohomology theories can be classified in the sense that there is an algebraic model for them. The model can be viewed as a category of sheaves over the space of subgroups of G. This has the character of a category of sheaves of modules over an algebraic variety we might call the spectrum of the sphere. The slides come from the talk I gave at the Saas conference in August, but the world-view behind them has undergone two very constructive upheavals since then.

Oct 11

Tue

Sarah Whitehouse (Sheffield)

Topology Seminar

16:00

Derived $A_{\infty}$ algebras and their homotopies

Hicks Seminar Room J11

Abstract:
The notion of a derived A-infinity algebra, due by Sagave, is a generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. I will describe a hierarchy of notions of homotopy between the morphisms of such algebras, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. Along the way, I'll give two new interpretations of derived A-infinity algebras. This is joint work with Joana Cirici, Daniela Egas Santander and Muriel Livernet

Oct 18

Tue

Dimitar Kodjabachev (Sheffield)

Topology Seminar

16:00

Gorenstein duality for topological modular forms with level structure.

Hicks Seminar Room J11

Oct 25

Tue

Luca Pol (Sheffield)

Topology Seminar

16:00

Connective K-theory from the global perspective

Hicks Seminar Room J11

Abstract:
In equivariant homotopy theory there are some theories that are defined in a uniform way for all groups in a specific class, rather than just for a particular group. The idea of global stable homotopy theory is to view this collection of compatible equivariant theories as one ``global'' object. One way to formalize this idea is to consider the well-known category of orthogonal spectra and to use a finer notion of equivalence: the global equivalence. In this talk, I will give an overview on global stable homotopy theory via orthogonal spectra and I will present a global equivariant version of connective topological K-theory. Time permitting, I will explain how to generalize this construction to obtain a global equivariant version of connective K-theory of C*-algebras.

Nov 1

Tue

Neil Strickland (Sheffield)

Topology Seminar

16:00

The known part of the Bousfield semiring

Hicks Seminar Room J11

Abstract:
The Bousfield semiring controls many interesting phenomena in stable homotopy theory. The literature contains many fragmentary results about the structure of this semiring. I will report on a project to combine all of these results into a single consolidated statement.

Nov 15

Tue

Dae Woong Lee (Chonbuk, Korea)

Topology Seminar

16:00

Strong homology, phantom maps, comultiplications and same n-types

Hicks Seminar Room J11

Abstract:
In this talk, the following topics in algebraic topology will be briefly outlined. (1) Strong (co)homology groups (2) Phantom maps (3) Comultiplications on a wedge of spheres (4) The same n-type structures of CW-complexes

Nov 22

Tue

Frank Neumann (Leicester)

Topology Seminar

16:00

Spectral sequences for Hochschild cohomology and graded centers of differential graded categories

Hicks Seminar Room J11

Abstract:
The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the possible failure of the characteristic homomorphism to be injective or surjective. To illustrate this, we will discuss several examples from geometry and topology, like modules over the dual numbers, coherent sheaves over algebraic curves, as well as examples related to free loop spaces and string topology. This is joint work with Markus Szymik (NTNU Trondheim).

Nov 29

Tue

Joao Faria Martins (Leeds)

Topology Seminar

16:00

Infinitesimal 2-braidings and KZ-2-connections.

Hicks Seminar Room J11

Abstract:
I will report on joint work with Lucio Cirio on categorifications of the Lie algebra of chord diagrams via infinitesimal 2-braidings in differential crossed modules.

Dec 6

Tue

Dean Barber (Sheffield)

Topology Seminar

16:00

A combinatorial model for Euclidean configuration spaces

Hicks Seminar Room J11

Abstract:
Configuration spaces appear in many areas of mathematics. They are simple to define but produce extremely complicated spaces. In this talk, we will introduce a family of posets, indexed by the natural numbers and finite sets, called the poset of chained linear preorders. It turns out that the geometric realisations of these posets are homotopy equivalent to configuration spaces on real vector spaces, and that the combinatorics involved can reveal some of the homotopical properties of these spaces.

Dec 13

Tue

Andrew Baker (University of Glasgow)

Topology Seminar

16:00

Hopf invariant one elements and E-infinity ring spectra

Hicks Seminar Room J11

Abstract:

At the prime 2, there are 4 Hopf invariant one elements (mod 2). These can be used to build some small complexes which also appear as low dimensional skeleta of some important classifying spaces and Thom spectra over them. Passing to free infinite loop spaces we can build some additional Thom spectra E-infinity ring spectra which have interesting properties. These have E-infinity ring maps to some important spectra including kO and tmf.

I will describe these spectra and some conjectures about splitting them and survey what is known so far.

Jan 17

Tue

Sara Kalisnik (Brown)

Topology Seminar

16:00

A short introduction to applied topology

Hicks Seminar Room J11

Abstract:
In the last two decades applied topologists have developed numerous methods for ‘measuring’ and building combinatorial representations of the shape of the data. The most famous example of the former is persistent homology and of the latter, mapper. I will briefly talk about both of these methods and show several successful applications. Time permitting I will talk about my work on making persistent homology easier to combine with standard machine learning tools.

Feb 7

Tue

Jeff Giansiracusa (Swansea)

Topology Seminar

16:00

Hicks Seminar Room J11

Feb 14

Tue

Nick Kuhn (Virginia)

Topology Seminar

16:00

The circle product of O-bimodules with O-algebras, with applications.

Hicks Seminar Room J11

Abstract:
If O is an operad (in a friendly category, e.g. the category of S-modules of stable homotopy theory), M is an O-bimodule, A is an O-algebra, then the circle product over O of M with A is again an O-algebra. A useful derived version is the bar construction B(M,O,A). We survey many interesting constructions on O-algebras that have this form. These include an augmentation ideal filtration of an augmented O-algebra A, the topological Andre-Quillen homology of A, the topological Hochschild homology of A, and the tensor product of A with a space. Right O-modules come with canonical increasing filtrations, and this leads to filtrations of all of the above. In particular, I can show that a filtration on TAQ(A) defined recently by Behrens and Rezk agrees with one I defined about a decade ago, as was suspected. This is joint work with Luis Pereira.

Feb 21

Tue

Angelica Osorno (Reed College)

Topology Seminar

16:00

On equivariant infinite loop space machines

Hicks Seminar Room J11

Abstract:
An equivariant infinite loop space machine is a functor that constructs genuine equivariant spectra out of simpler categorical or space level data. In the late 80's Lewis-May-Steinberger and Shimakawa developed generalizations of the operadic approach and the Gamma-space approach respectively. In this talk I will describe work in progress that aims to understand these machines conceptually, relate them to each other, and develop new machines that are more suitable for certain kinds of input. This work is joint with Anna Marie Bohmann, Bert Guillou, Peter May and Mona Merling.

Feb 28

Tue

Gareth Williams (Open)

Topology Seminar

16:00

Weighted projective spaces, equivariant K-theory and piecewise algebra

Hicks Seminar Room J11

Abstract:
Weighted projective spaces are interesting through many lenses: for example, as natural generalisations of ordinary projective spaces, as toric varieties and as orbifolds. From the point of view of algebraic topology, it is natural to study their algebraic topological invariants – notably, their (equivariant) cohomology rings. Recent work has provided satisfying qualitative descriptions for these rings, in terms of piecewise algebra, for various cohomology theories. This talk will introduce weighted projective spaces as toric varieties and survey results on their (equivariant) cohomology rings, with particular focus on equivariant K-theory. It will conclude with recent results of Megumi Harada, Tara Holm, Nige Ray and the speaker, and indicate the flavour of current work of Tara Holm and the speaker.

Mar 7

Tue

Will Mycroft

Topology Seminar

16:00

Plethories of Cohomology Operations

Hicks Seminar Room J11

Abstract:
Cohomology operations are a very useful property of a cohomology theory. The collection of cohomology operations has a very rich structure. Historically the dual notion, of homology cooperations, have been the main target of attention and a nice algebraic structure called a Hopf ring has been used to understand these. Unfortunately, the Hopf ring contains no structure that is dual to the notion of composition. Boardman, Wilson and Johnson attempt to rectify this situation by defining an enriched Hopf ring, although this structure is rather less pleasant. A 2009 theorem of Stacey and Whitehouse shows that the collection of cohomology operations has the structure of an algebraic object called a plethory and this expresses all the structure, including composition. In this talk I shall define the above concepts and illustrate some examples of plethories for known cohomology theories.

Mar 14

Tue

Dimitar Kodjabachev (Sheffield)

Topology Seminar

16:00

Hicks Seminar Room J11

Mar 14

Tue

Dimitar Kodjabachev (Sheffield)

Topology Seminar

16:00

Gorenstein duality for topological modular forms with level structure

Hicks Seminar Room J11

Abstract:
Gorenstein duality is a homotopy theoretic framework that allows one to view a number of dualities in algebra, geometry and topology as examples of a single phenomenon. I will briefly introduce the framework and concentrate on illustrating it with examples coming from derived algebraic geometry, especially topological modular forms with level structure.

Apr 25

Tue

Ana Lecuona

Topology Seminar

16:00

Complexity and Casson-Gordon invariants

Hicks Seminar Room J11

Abstract:
Homology groups provide bounds on the minimal number of handles needed in any handle decomposition of a manifold. We will use Casson-Gordon invariants to get better bounds in the case of 4-dimensional rational homology balls whose boundary is a given rational homology 3-sphere. This analysis can be used to understand the complexity of the discs associated to ribbon knots in S^3. This is a joint work with P. Aceto and M. Golla.

May 2

Tue

John Greenlees (Sheffield)

Topology Seminar

16:00

Thick and localizing subcategories of rational G-spectra

Hicks Seminar Room J11

Abstract:
The Balmer spectrum of the category of rational G-spectra as a poset is the closed subgroups of G under cotoral inclusion. In December, I posted a preprint on the arXiv that proved this for tori: the talk will describe a much simpler proof of a theorem for all compact Lie groups. The method applies in other contexts with only a few special inputs from equivariant topology: the Localization Theorem, The calculation of the Burnside ring and a method of calculation for maps between free G-spectra.

May 16

Tue

Sarah Browne (Sheffield)

Topology Seminar

00:00

An orthogonal quasi-spectrum for graded E-theory

Hicks Seminar Room J11

Abstract:
Graded E-theory is a bivariant functor from the category where objects are graded C*-algebras and arrows are graded *-homomorphisms to the category where objects are abelian groups and arrows are group homomorphisms. It is bivariant in the sense that it is a cohomology theory in its first variable and a homology theory in its second variable. In this talk I'll give a description of a quasi-topological space and explain why this notion is necessary in our case. We will define the notion of an orthogonal quasi-spectrum as an orthogonal spectrum for quasi-topological spaces, and further give the quasi-topological spaces to form the spectrum for graded E-theory. If time allows I will give the smash product structure.

May 16

Tue

Sarah Browne (Sheffield)

Topology Seminar

16:00

Quasi-topological assembly for K theory

Hicks Seminar Room J11

May 23

Tue

Magdalena Kedziorek (Lausanne)

Topology Seminar

16:00

Rational commutative ring G-spectra

Hicks Seminar Room J11

Abstract:
Recently, there has been some new understanding of various possible commutative ring G-spectra. In this talk I will recall these possibilities and discuss the most naive (or trivial) commutative ring G-spectra. Then I will sketch the main ingredients coming into the proof that if G is finite and we work rationally these objects correspond to (the usual) commutative differential algebras in the algebraic model for rational G-spectra. This is joint work with David Barnes and John Greenlees.

Jun 6

Tue

Titanic Ten

Topology Seminar

16:00

Gong Show

Hicks Seminar Room J11

Abstract:
Tea then ten ten-minute talks.

Oct 5

Thu

Simon Willerton (Sheffield)

Topology Seminar

16:00

The magnitude of odd balls

Hicks Seminar Room J11

Abstract:
Tom Leinster introduced the magnitude of finite metric spaces by formal analogy with his notion of Euler characteristic of finite categories. This can be thought of an 'effective number of points' n the metric space. It soon became clear that this notion of magnitude could

Oct 12

Thu

Akos Matszangosz

Topology Seminar

16:00

Real enumerative geometry and equivariant cohomology: Borel-Haefliger type theorems

Hicks Seminar Room J11

Abstract:
Enumerative geometry studies questions of the type: how many geometric objects satisfy a prescribed set of (generic) conditions? Over the complex field the answer is a single number. However, over R the answer depends on the configuration. A theorem of Borel and Haefliger states that mod 2 the answer is the same. Thom realized, that for a generic a) smooth, b) holomorphic map f, the cohomology class [Si(f)] of the singular points of f of a given type can be expressed as a universal polynomial evaluated at the characteristic classes of the map. The second theorem of Borel and Haefliger states that mod 2, the universal polynomial is the same in the smooth and holomorphic case. In this talk I plan to discuss these questions from the point of view of equivariant topology. The spaces satisfying the condition of the Borel-Haefliger theorem are part of a class of Z2-spaces called conjugation spaces introduced by Hausmann, Holm and Puppe. Analogously we introduce a class of U(1)-spaces which we call circle spaces in an attempt to say something more than parity about these questions. This is joint work with László Fehér.

Oct 26

Thu

Scott Balchin (Sheffield)

Topology Seminar

16:00

Lifting cyclic model structures to the category of groupoids

Hicks Seminar Room J11

Abstract:
Abstract: We consider the problem of lifting certain Quillen model structures on the category of cyclic sets to the category of groupoids, echoing the construction of the Thomason model structure on Cat. We prove that this model structure only captures the theory of homotopy 1-types, and as a consequence, that SO(2)-equivariant homotopy 1-types cannot be encoded in a discrete manner. We will fully describe all of the components required for this model structure, in particular, assuming no familiarity with the model structures on cyclic sets or the Thomason model structure on Cat. This work is joint with Richard Garner.

Nov 2

Thu

Julian Holstein (Lancaster)

Topology Seminar

16:00

Maurer-Cartan elements and infinity local systems

Hicks Seminar Room J11

Abstract:
Maurer-Cartan elements for differential graded Lie algebras or associative algebras play an important role in several branches of mathematics, in particular for classifying deformations . There are different sensible notions of equivalence for Maurer-Cartan elements, and while they agree in the nilpotent case, the general theory is not yet well-understood. This talk will compare gauge equivalence and different notions of homotopy equivalence for Maurer-Cartan elements of a dg-algebra. As an application we extend the study of cohesive modules introduced by Block, and find a new algebraic characterisation of infinity local systems on a topological space. This is joint work with Joe Chuang and Andrey Lazarev.

Nov 9

Thu

Constanze Roitzheim (Kent)

Topology Seminar

16:00

K-local equivariant rigidity

Hicks Seminar Room J11

Abstract:
Equivariant stable homotopy concerns the study of objects with symmetry. It has been shown recently by Patchkoria that the G-equivariant stable homotopy category is uniquely determined by its triangulated structure, G-action and induction/transfer/restriction maps. In particular this implies that all reasonable categories of G-spectra realise the same homotopy theory. We consider this result with respect to equivariant K-theory, which merges model category techniques, equivariant structures and calculations from the stable homotopy groups of spheres.

Nov 16

Thu

Markus Hausmann (Copenhagen)

Topology Seminar

16:00

The Balmer spectrum of the equivariant homotopy category of a finite abelian group

Hicks Seminar Room J11

Abstract:
One of the basic tools to study a tensor-triangulated category is a classification of its thick tensor ideals. In my talk, I will discuss such a classification for the category of compact G-spectra for a finite abelian group G. This is joint work with Tobias Barthel, Niko Naumann, Thomas Nikolaus, Justin Noel and Nat Stapleton, and builds on work of Strickland and Balmer-Sanders.

Nov 23

Thu

Claudia Scheimbauer (Oxford)

Topology Seminar

16:00

Fully extended functorial field theories and dualizability in the higher Morita category

Hicks Seminar Room J11

Abstract:
Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic. Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms. After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualizablity in a ``factorization version’’ of the Morita category and using this, examples of low-dimensional field theories “relative” to their observables. An example will be given by polynomial differential operators, i.e. the Weyl algebra, in positive characteristic and its center. This is joint work with Owen Gwilliam.

Nov 30

Thu

Neil Strickland (Sheffield)

Topology Seminar

16:00

Thoughts on the Telescope Conjecture

Hicks Seminar Room J11

Abstract:
Ravenel's 1984 paper "Localization with respect to certain periodic theories" posed a series of highly prescient conjectures, most of which were later proved by Hopkins, Devinatz and Smith. These results form the heart of chromatic homotopy theory. One conjecture, called the Telescope Conjecture, remained unproven. It can be formulated in many ways, one of which is as follows: if $X$ is a spectrum such that $v_n^{-1}X$ is defined, and $BP_*(v_n^{-1}X)=0$, then already $v_n^{-1}X=0$. This is trivial for $n=0$, and is true for $n=1$ by a theorem of Miller. However, many people including Ravenel came to believe that it is probably false for $n\geq 2$. In 2000 Mahowald, Ravenel and Shick published a paper describing their attempt to disprove the conjecture. They constructed a certain spectral sequence, and showed that the conjecture would imply properties of the spectral sequence that they found implausible, but they were not able to complete the proof of impossibility.

This talk will survey this work, and present some small new ideas about properties of certain spectra $T(n,q)$ that play an important role here and in some related areas.

Dec 14

Thu

Danny Sugrue (Queens University Belfast)

Topology Seminar

16:00

The title is Rational Mackey functors of profinite groups.

Hicks Seminar Room J11

Abstract:
Rational Mackey functors for a compact topological group G are a useful tool for modelling rational G equivariant cohomology theories. Having a better understanding of Mackey functors will enhance our understanding of G-cohomology theories and G-equivariant homotopy theory in general. In the compact Lie group case, rational Mackey functors have been studied extensively by John Greenlees (and others). In this talk we will discuss what can be shown in the case where G is profinite (an inverse limit of finite groups).

Feb 15

Thu

David Barnes (Queen's University Belfast)

Topology Seminar

16:00

Cohomological dimension of profinite spaces

Hicks Seminar Room J11

Abstract:
I will introduce the notion of rational cohomological dimension of topological spaces and show a simple way to calculate it when we restrict ourselves to a certain class of topological spaces. Very roughly, the r.c.d of a space X is the largest p such that the pth rational cohomology of X is non-zero. This invariant can be calculated in terms of the more geometric notion of sheaves on X. The category of sheaves on X is an abelian category and the injective dimension of this category is the r.c.d of X. This is a standard way to calculate the the r.c.d. of a space, but can be rather difficult. In this talk, I will describe how for profinite spaces, this injective dimension is related to a simpler notion: the Cantor-Bendixson dimension of the space. There will be a number of pictures and some nice examples illustrating the calculations.

Feb 22

Thu

Luca Pol (Sheffield)

Topology Seminar

16:00

On the geometric isotropy of a compact rational global spectrum

Hicks Seminar Room J11

Abstract:
In this talk I will explain a way to detect groups in the geometric isotropy of a compact rational global spectrum. As an application, I will show that the Balmer spectrum of the rational global stable homotopy category exhibits at least two different types of prime: group and multiplicative primes.

Mar 1

Thu

Christian Wimmer (Bonn)

Topology Seminar

16:00

A model for equivariant commutative ring spectra away from the group order

Hicks Seminar Room J11

Abstract:
Stable homotopy theory simplifies drastically if one consider spectra up to rational equivalence. It is a classical result that taking homotopy groups induces an equivalence $$G \text{-} \mathcal{SHC} \simeq_{\mathbb{Q}} \text{gr.} \prod_{(H \leq G)} \mathbb{Q} [WH] \text{-mod}$$ between the genuine $G$-equivariant stable homotopy category ($G$ finite) and the category of graded modules over the Weyl groups $WH$ indexed by the conjugacy classes of subgroups of $G$. However, this approach is too primitive to be useful for the comparison of highly structured ring spectra in this setting.

Let $R \subset \mathbb{Q}$ be a subring such that $|G|$ is invertible in $R$. I will explain how geometric fixed points equipped with additional norm maps related to the Hill-Hopkins-Ravenel norms can be used to give an $R$-local model: They induce an equivalence $$\text{Com}(G\text{-Sp}) \simeq_R \text{Orb}_G \text{-Com}(\text{Sp})$$ between the $R$-local homotopy theories of genuine commutative $G$-ring spectra and $\text{Orb}_G$-diagrams in non-equivariant commutative ring spectra, where $\text{Orb}_G$ is the orbit category of the group $G$. As a corollary this gives an algebraic model $$\text{Com}(G\text{-Sp})_\mathbb{Q} \simeq \text{Orb}_G \text{-CDGA}_\mathbb{Q}$$ for rational ring spectra in terms of commutative differential algebras. I will also try to indicate the analogous global equivariant statements.

Mar 15

Thu

Simon Wood (Cardiff)

Topology Seminar

16:00

Questions in representation theory inspired by conformal field theory

Hicks Seminar Room J11

Abstract:
Two dimensional conformal field theories (CFTs) are conformally invariant quantum field theories on a two dimensional manifold. What distinguishes two dimensions from all others is that the (Lie) algebra of local conformal transformations become infinite dimensional. This extraordinary amount of symmetry allows certain conformal field theories to be solved by symmetry considerations alone. The most intensely studied type of CFT, called a rational CFT, is characterised by the fact that its representation theory is completely reducible and that there are only a finite number isomorphism classes of irreducibles. The representation categories of these CFTs form so called modular tensor categories which have important applications in the construction of 3-manifold invariants. In this talk I will discuss recent attempts at generalising this very rich structure to CFTs whose representation categories are neither completely reducible nor finite.

Mar 26

Mon

Hans Werner Henn (Strasbourg)

Topology Seminar

16:00

The centralizer resolution of the K(2)-local sphere at the prime 2.

Hicks Seminar Room J11

Abstract:
In the last few years two different resolutions of the K(2)-local sphere at the prime 3 have been used very successfully to settle some basic problems in K(2)-local stable homotopy theory like the chromatic splitting conjecture, the calculation of Hopkins' K(2)-local Picard group and determining $K(2)-local Brown-Comentz duality. The focus is now moving towards the prime 2 where one can hope for similar progress. In this talk we concentrate on one of these two resolutions, the centralizer resolution at the prime 2.

Oct 18

Thu

Simon Willerton (Sheffield)

Topology Seminar

16:00

The Legendre-Fenchel transform from a category theoretic perspective

Hicks Seminar Room J11

Abstract:
The Legendre-Fenchel transform is a classical piece of mathematics with many applications. In this talk I'll show how it arises in the context of category theory using categories enriched over the extended real numbers $\overline{ \mathbb{R}}:=[-\infty,+\infty]$. It turns out that it arises out of nothing more than the pairing between a vector space and its dual in the same way that the many classical dualities (eg. in Galois theory or algebraic geometry) arise from a relation between sets.
I will assume no knowledge of the Legendre-Fenchel transform and no knowledge of enriched categories.

Oct 25

Thu

Anna Marie Bohmann (Vanderbilt)

Topology Seminar

16:00

Graded Tambara Functors

Hicks Seminar Room J11

Abstract:
Let G be a finite group. The coefficients of G-equivariant cohomology theories naturally form a type of structure called a Mackey functor, which incorporates data coming from each subgroup of G. When the cohomology theory is a G-ring commutative spectrum---meaning that is has an equivariant multiplication---interesting new structures arise. In particular, work of Brun and of Strickland shows that the zeroth homotopy groups have norm maps which yield the structure of a Tambara functor. In this talk, I discuss joint work with Vigleik Angeltveit on the algebraic structure induced by norm maps on the higher homotopy groups, which we call a graded Tambara functor.

Nov 1

Thu

Markus Szymik (NTNU)

Topology Seminar

16:00

Quandles, knots, and homotopical algebra

Hicks Seminar Room J11

Abstract:
Knots and their groups are a traditional topic of geometric topology. In this talk I will explain how aspects of the subject can be approached using ideas from Quillen’s homotopical algebra, rephrasing old results and leading to new ones.

Nov 22

Thu

Robert Bruner (Wayne State)

Topology Seminar

16:00

The mod 2 Adams Spectral Sequence for Topological Modular Forms

Hicks Seminar Room J11

Abstract:
In joint work with John Rognes, we have computed the 2-local homotopy of tmf, with full details. We first compute the cohomology of A(2) by a method of general interest. Grobner bases play a key role in allowing us to give a useful description it. I will briefly describe this. We then show that all the Adams spectral sequence differentials follow from general properties together with three key relations in the homotopy of spheres. We then compute the hidden extensions and the relations in homotopy using the cofibers of 2, eta and nu. This allows us to give a clear and memorable description of tmf_*. I will end with a brief description of the duality present in tmf_* coming from the Anderson duality for tmf.

Dec 18

Tue

Alexis Virelizier (Lille)

Topology Seminar

16:00

Generalized Kuperberg invariants of 3-manifolds

Hicks Seminar Room J11

Abstract:
In the 90s, Kuperberg defined a scalar invariant of 3-manifolds from each finite-dimensional involutory Hopf algebra over a field. The construction is based on the presentation of 3-manifolds by Heegaard diagrams and involves tensor products of the structure tensors of the Hopf algebra. These tensor products are then contracted using integrals of the Hopf algebra to obtain the scalar invariant. We generalize this construction by contracting the tensor products with other morphisms. Examples of such morphisms are derived from involutory Hopf algebras in symmetric monoidal categories. This is a joint work with R. Kashaev.

Feb 14

Thu

Andrey Lazarev (Lancaster)

Topology Seminar

16:00

Homotopy theory of monoids

Hicks Seminar Room J11

Abstract:
I will explain how the category of discrete monoids models the homotopy category of connected spaces. This correspondence is based on derived localization of associative algebras and could be viewed as an algebraization result, somewhat similar to rational homotopy theory (although not as structured). Closely related to this circle of ideas is a generalization of Adams’s cobar construction to general nonsimply connected spaces due to recent works of Rivera-Zeinalian and Hess-Tonks. (joint with J. Chuang and J. Holstein)

Feb 20

Wed

Clark Barwick (Edinburgh)

Topology Seminar

16:00

Primes, knots, and exodromy

LT11

Abstract:
Half a century ago, Barry Mazur and David Mumford suggested a remarkable dictionary between prime numbers and knots. I will explain how the story of exodromy permits one to make this dictionary precise, and I will describe some applications.

Feb 28

Thu

Scott Balchin (Warwick)

Topology Seminar

16:00

Adelic reconstruction in prismatic chromatic homotopy theory

Hicks Seminar Room J11

Abstract:
Prismatic homotopy theory is the study of stable monoidal homotopy theories through their Balmer spectrum. In this talk, I will discuss how one can use localised p-complete data at each Balmer prime in an adelic fashion to reconstruct the homotopy theory in question. There are two such models, one is done by moving to categories of modules, which, for example, recovers the algebraic models for G-equivariant cohomology theories. The other, newer model, works purely at the categorical level and requires the theory of weighted homotopy limits. This is joint work with J.P.C Greenlees.

Mar 7

Thu

Irakli Patchkoria (Aberdeen)

Topology Seminar

16:00

Computations in real topological Hochschild and cyclic homology

Hicks Seminar Room J11

Abstract:
The real topological Hochschild and cyclic homology (THR, TCR) are invariants for rings with anti-involution which approximate the real algebraic K-theory. In this talk we will introduce these objects and report about recent computations. In particular we will dicuss components of THR and TCR and some recent and ongoing computations for finite fields. This is all joint with E. Dotto and K. Moi.

Mar 14

Thu

Neil Strickland (Sheffield)

Topology Seminar

16:00

Dilation of formal groups, and potential applications

Hicks Seminar Room J11

Abstract:
I will describe an extremely easy construction with formal group laws, and a slightly more subtle argument to show that it can be done in a coordinate-free way with formal groups. I will then describe connections with a range of other phenomena in stable homotopy theory, although I still have many more questions than answers about these. In particular, this should illuminate the relationship between the Lambda algebra and the Dyer-Lashof algebra at the prime 2, and possibly suggest better ways to think about related things at odd primes. The Morava K-theory of symmetric groups is well-understood if we quotient out by transfers, but somewhat mysterious if we do not pass to that quotient; there are some suggestions that dilation will again be a key ingredient in resolving this. The ring $MU_*(\Omega^2S^3)$ is another object for which we have quite a lot of information but it seems likely that important ideas are missing; dilation may also be relevant here.

Slides

Mar 21

Thu

Mike Prest (Manchester)

Topology Seminar

16:00

Categories of imaginaries for additive categories

Hicks Seminar Room J11

Abstract:
There is a construction of Freyd which associates, to any ring R, the free abelian category on R. That abelian category may be realised as the category of finitely presented functors on finitely presented R-modules. It has an alternative interpretation as the category of (model-theoretic) imaginaries for the category of R-modules. In fact, this extends to additive categories much more general than module categories, in particular to finitely accessible categories with products and to compactly generated triangulated categories. I will describe this and give some examples of its applications.

Mar 28

Thu

Jordan Williamson (Sheffield)

Topology Seminar

16:00

A Left Localization Principle and Cofree G-Spectra

Hicks Seminar Room J11

Abstract:
Greenlees-Shipley developed a Cellularization Principle for Quillen adjunctions in order to attack the problem of constructing algebraic models for rational G-spectra. One example of this was the classification of free rational G-spectra as torsion modules over the cohomology ring H*(BG) (for G connected). This has some disadvantages; namely that it is not monoidal and that torsion modules supports only an injective model structure. I will explain a related method called the Left Localization Principle, and how this can be used to construct a monoidal algebraic model for cofree G-spectra. This will require a tour through the different kinds of completions available in homotopy theory. This is joint work with Luca Pol.

Apr 4

Thu

Richard Hepworth (Aberdeen)

Topology Seminar

16:00

CANCELLED

Hicks Seminar Room J11

May 2

Thu

Celeste Damiani (Leeds)

Topology Seminar

16:00

TBA

Hicks Seminar Room J11

May 17

Fri

Gong Show

Topology Seminar

16:00

Hicks Seminar Room J11

Oct 3

Thu

Ulrich Pennig (Cardiff)

Topology Seminar

16:00

Equivariant higher twisted K-theory of SU(n) via exponential functors

Hicks Seminar Room J11

Abstract:
Twisted K-theory is a variant of topological K-theory that allows local coefficient systems called twists. For spaces and twists equipped with an action by a group, equivariant twisted K-theory provides an even finer invariant. Equivariant twists over Lie groups gained increasing importance in the subject due to a result by Freed, Hopkins and Teleman that relates the corresponding K-groups to the Verlinde ring of the associated loop group. From the point of view of homotopy theory only a small subgroup of all possible twists is considered in classical treatments of twisted K-theory. In this talk I will discuss an operator-algebraic model for equivariant higher (i.e. non-classical) twists over SU(n) induced by exponential functors on the category of vector spaces and isomorphisms. These twists are represented by Fell bundles and the C*-algebraic picture allows a full computation of the associated K-groups at least in low dimensions. I will also draw some parallels of our results with the FHT theorem. This is joint work with D. Evans.

Oct 10

Thu

Daniel Graves (Sheffield)

Topology Seminar

16:00

Now that's what I call...homology theories for algebras

Hicks Seminar Room J11

Abstract:
Homology theory for algebras was first introduced by Hochschild in the 40s to classify extensions of associative algebras. Since then a great many homology theories have been introduced to encode and detect desirable properties of algebras. I will describe a selection of these homology theories, discuss how they relate to one another and introduce some chain complexes for computing them.

Oct 17

Thu

Alexander Schenkel (Nottingham)

Topology Seminar

16:00

Higher categorical structures in algebraic quantum field theory

Hicks Seminar Room J11

Abstract:
Algebraic quantum field theory (AQFT) is a well-established framework to axiomatize and study quantum field theories on Lorentzian manifolds, i.e. spacetimes in the sense of Einstein’s theory of general relativity. In the first part of the talk, I will try to explain both the physical context and the mathematical formalism of AQFT in a way that is hopefully of interest to topologists. In the second part of the talk, I will give an overview of our recent works towards establishing a higher categorical framework for AQFT. This will include the construction of examples of such higher categorical theories from (linear approximations of) derived stacks and a discussion of their descent properties.

Oct 24

Thu

Richard Hepworth (Aberdeen)

Topology Seminar

16:00

Homological Stability: Coxeter, Artin, Iawahori-Hecke

Hicks Seminar Room J11

Abstract:
Homological stability is a topological property that is satisfied by many families of groups, including the symmetric groups, braid groups, general linear groups, mapping class groups and more; it has been studied since the 1950's, with a lot of current activity and new techniques. In this talk I will explain a set of homological stability results from the past few years, on Coxeter groups, Artin groups, and Iwahori-Hecke algebras (some due to myself and others due to Rachael Boyd). I won't assume any knowledge of these things in advance, and I will try to introduce and motivate it all gently!

Oct 31

Thu

Ai Guan (Lancaster)

Topology Seminar

16:00

A model structure of second kind on differential graded modules

Hicks Seminar Room J11

Abstract:
Koszul duality is a phenomenon appearing in many areas of mathematics, such as rational homotopy theory and deformation theory. For differential graded (dg) algebras, the modern formulation of Koszul duality says there is a Quillen equivalence between model categories of augmented dg algebras and conilpotent dg coalgebras, and also Quillen equivalences between corresponding dg modules/comodules. I will give an overview of this circle of ideas, and then consider what happens when the conilpotence condition is removed. The answer to this question leads to an exotic model structure on dg modules that is "of second kind", i.e. weak equivalences are finer than quasi-isomorphisms. This is based on joint work with Andrey Lazarev from the recent preprint https://arxiv.org/abs/1909.11399.

Nov 7

Thu

Emanuele Dotto (Warwick)

Topology Seminar

16:00

The Witt vectors with coefficients

Hicks Seminar Room J11

Abstract:
We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.

Nov 14

Thu

Greg Stevenson (Glasgow)

Topology Seminar

16:00

An introduction to derived singularities

LT7

Abstract:
The aim of this talk is to give an introduction to what it might mean for a differential graded algebra (or ring spectrum) to be singular, in a sense analogous to the situation in algebraic geometry. As in geometry one can distinguish between smoothness and regularity, and I'll discuss both concepts and their relationship. The failure of the latter, i.e. the presence of singularities, can in good situations be described by a corresponding singularity category and time permitting I'll sketch how this category can be defined as in joint work with John Greenlees.

Nov 21

Thu

Abigail Linton (Southampton)

Topology Seminar

16:00

Non-trivial Massey products in moment-angle complexes

Hicks Seminar Room J11

Abstract:
A moment-angle complex $\mathcal{Z}_\mathcal{K}$ is obtained by associating a product of discs and circles to each simplex in a simplicial complex $\mathcal{K}$ and gluing these products according to how the corresponding simplices intersect. These spaces can have a complicated topological structure. For example, Baskakov (2003) found examples of non-trivial Massey products in the cohomology of moment-angle complexes. I will give a complete combinatorial classification of lowest-degree non-trivial triple Massey products in the cohomology of moment-angle complexes and describe constructions of simplicial complexes that give non-trivial higher Massey products on classes of any degree.

Dec 5

Thu

Ieke Moerdijk (Utrecht/Sheffield)

Topology Seminar

16:00

Labelled configuration spaces and a theorem of Segal

Hicks Seminar Room J11

Abstract:
As a digression from (and sufficiently independently of) the course on configuration spaces, I will explain Graeme Segal's proof that configuration spaces with labels in a pointed space $X$ model $\Omega^n \Sigma^n X$.

Dec 12

Thu

Gong Show

Topology Seminar

16:00

Hicks Seminar Room J11

Feb 13

Thu

Severin Bunk (Hamburg University)

Topology Seminar

16:00

Smooth Open-Closed Functorial Field Theories from B-Fields and D-Branes

Hicks Seminar Room J11

Abstract:
Bundle gerbes are a categorification of line bundles, and their connections model the B-field in string theory. In this talk we show how bundle gerbes with connection and their D-branes give rise to smooth open-closed field theories (OCFFTs) on a manifold M in a functorial manner. The key ingredients for this construction are the 2-categorical structure of bundle gerbes, the transgression of gerbes and D-branes to spaces of loops and paths in M, as well as a formalisation of the Wess-Zumino amplitude on surfaces with corners. After giving an overview of these concepts, we will explain how they combine to yield the desired smooth OCFFTs on M. This is based on an ongoing collaboration with Konrad Waldorf.

Feb 20

Thu

Niall Taggart (Queen's University Belfast)

Topology Seminar

16:00

Comparing functor calculi

Hicks Seminar Room J11

Abstract:
Functor calculus is a categorification of Taylor's Theorem from differential calculus. Given a functor, one can assign a sequence of polynomial approximations, which assemble into a Taylor tower, similar to the Taylor series from differential calculus. In this talk, I will introduce several variants of functor calculus together with their associated model categories, and demonstrate how one may compare these calculi both on a point-set and model categorical level.

Feb 27

Thu

Ai Guan (Lancaster)

Topology Seminar

16:00

Koszul duality for derived categories of second kind

Hicks Seminar Room J11

Abstract:
Koszul duality is a phenomenon appearing in many areas of mathematics, such as rational homotopy theory and deformation theory. For differential graded (dg) algebras, the modern formulation of Koszul duality says there is a Quillen equivalence between model categories of dg algebras and conilpotent dg coalgebras, and their corresponding dg modules/comodules. I will give an overview of this circle of ideas, and then consider what happens when the conilpotence condition is removed. The answer to this question leads to an exotic model structure on dg modules that is "of second kind", i.e. weak equivalences are finer than quasi-isomorphisms. This is joint work with Andrey Lazarev based on the preprint https://arxiv.org/abs/1909.11399.

Apr 2

Thu

Nicola Bellumat (University of Sheffield)

Topology Seminar

16:00

Iterated chromatic localization

Abstract:
The work of Ravanel, Devinatz, Hopkins and Smith in the Eighties provided the basis of chromatic homotopy theory: its protagonists are the Morava theories E(n) and K(n), whose associated Bousfield localizations provide optimal means to decompose the stable homotopy category. It comes naturally to wonder how the compositions of such localizations behave: there are classical results regarding the relationship of the Bousfield classes of wedges of the above spectra which lead us to expect some kind of regularity. In this talk I will present a joint work with N. Strickland which provides a positive result in this direction: we show that, fixed an upper bound n for the chromatic height, the compositions of localizations with respect to spectra which are wedges of K(i), for i lesser or equal n, are only finitely many up to isomorphism. We formulated our proof in the language of derivators, thus I will provide a brief overview.

Apr 16

Thu

Jocelyne Ishak (Vanderbilt University)

Topology Seminar

16:00

The naive commutative structure on rational equivariant $K$-theory

Abstract:
Modeling rational spectra via algebraic data has a long and fruitful history in homotopy theory. More precisely, rational spectra are equivalent to rational chain complexes, and this algebraic data is called an algebraic model for rational spectra. Our goal is to understand rational equivariant $K$-theory as a naive commutative ring spectrum when G is a finite group. We do this by calculating its image in the algebraic model for naive-commutative ring G-spectra given by Barnes, Greenlees and Kędziorek. Our calculations show that these spectra are unique as naive-commutative ring spectra in the sense that they are determined up to weak equivalence by their homotopy groups. This work is joint with Anna Marie Bohmann, Christy Hazel, Magdalena Kędziorek, and Clover May

Apr 23

Thu

Liran Shaul (Charles University in Prague)

Topology Seminar

16:00

The Cohen-Macaulay property in derived algebraic geometry

Abstract:
In this talk we explain how to extend the theory of Cohen-Macaulay rings and Cohen-Macaulay modules to the setting of commutative DG-rings. We will explain how by studying local cohomology in the DG-setting, one obtains certain amplitude inequalities about certain DG-modules of finite injective dimension. When these inequalities are equalities, we arrive to the notion of a Cohen-Macaulay DG-ring. We show that these arise naturally in many situations, and explain their basic theory. We then explain that this situation is the generic local situation in derived algebraic geometry; under mild hypothesis, every eventually coconnective locally noetherian derived scheme is Cohen-Macaulay on a dense open set.

Apr 30

Thu

Magdalena Kedziorek (Radboud University Nijmegen)

Topology Seminar

16:00

Genuine commutative structure on rational equivariant K-theory

meet.google.com/sqb-gwhq-dgk

Abstract:
In a recent talk at this seminar Jocelyne Ishak described a proof that rational equivariant K-theory admits a unique naive-commutative structure when the group of equivariance is finite abelian. A natural question to ask is what can we say about other levels of commutative structures on rational equivariant K-theory? Using the result of Wimmer which provides an algebraic model for rational genuine commutative ring G-spectra when G is a finite group I will sketch a proof that rational equivariant K-theory has a unique genuine-commutative ring structure for some groups G. This is work in progress with Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak and Clover May. I will start by recalling necessary results mentioned by Jocelyne in her talk and give a short introduction to different levels of commutativity present in the equivariant world.

May 7

Thu

Peter Symonds (Manchester)

Topology Seminar

16:00

Rank, Coclass and Cohomology.

meet.google.com/hdp-cfvn-wak

Abstract:
We prove that for any prime p the finite p-groups of fixed coclass have only finitely many different mod-p cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of fixed rank first conjectured by Diaz, Garaialde and Gonzalez.

May 14

Thu

Sarah Whitehouse (Sheffield)

Topology Seminar

16:00

Multicomplexes and their homotopy theory

meet.google.com/hdp-cfvn-wak

Abstract:
A multicomplex is an algebraic structure generalizing the notion of a (graded) chain complex and that of a bicomplex. The structure involves a family of higher “diﬀerentials” indexed by the non-negative integers. The terms twisted chain complex and D-infinity-module are also used. Multicomplexes have arisen in many diﬀerent places and play an important role in homotopical and homological algebra. I'll try to survey some of this landscape and talk about joint work with Xin Fu, Ai Guan and Muriel Livernet giving a family of model category structures on multicomplexes.

May 21

Thu

Gong Show

Topology Seminar

16:00

Hicks Seminar Room J11

Oct 6

Thu

Markus Szymik (Sheffield)

Topology Seminar

16:00

Work in progress on knots and primes

F38

Abstract:
Analogies between low-dimensional topology and number theory have been suggested for over a century. One thing I am interested in at the moment is seeing how we can use the algebra of racks and quandles to classify such objects and understand their symmetries. In this talk, I will briefly introduce this algebra, sketch my work in progress, and indicate some possible future directions if time permits.

Oct 13

Thu

Daniel Graves (Leeds)

Topology Seminar

16:00

A talk on the PROBlem of PROducing PROPer indexing categories for categories of monoids

Hicks Seminar Room J11

Abstract:
PROPs are "product and permutation categories". They encode structure borne by objects in a symmetric monoidal category. In this talk I will discuss how the PROP that indexes the structure of a monoid in a symmetric monoidal category is closely related to the theory of crossed simplicial groups. I will then report on recent work (and work in progress) which generalizes this in two ways. I will discuss, firstly, how we can extend known results in the symmetric case to cover monoids with extra structure and, secondly, how we can translate all the results to the setting of braided monoidal categories.

Oct 27

Thu

Paul Mitchener (Sheffield)

Topology Seminar

16:00

Assembly Maps

Hicks Seminar Room J11

Abstract:
An assembly map is a universal approximation of a homotopy-invariant functor by a generalised homology. In this talk, we introduce the concept and examine examples. When we have an assembly map, we have an associated generalised Novikov conjecture, stating that the map is injective when applied to the classifying space of a group. The plan is to show a general technique coming from coarse geometry to prove injectivity of the assembly map for certain classes of groups.

Nov 3

Thu

Simon Willerton (Sheffield)

Topology Seminar

16:00

Metric spaces, enriched categories and convexity

Hicks Seminar Room J11

Abstract:
The notion of convexity of sets can be captured in a category theoretic way using a what is known as a monad which associates to a space the finite formal convex combinations of elements. Various authors have looked at such convexity monads on categories of metric spaces. It became clear to me that the work of Fritz-Perrone on this could be naturally expressed if you considered metric spaces as enriched categories, that is categories enriched over a category non-negative real numbers. In this talk I'll explain this point of view and how notions of concave and convex maps naturally arise when you think higher-categorically. The work is motivated by an attempt to combine two categorical approaches to thermodynamics, one from Lawvere involving enriched categories and one from Baez-Lynch-Moeller involving convexity; I might mention some aspects of that if time permits.

Nov 17

Thu

Sarah Whitehouse (Sheffield)

Topology Seminar

16:00

Homotopy theory of spectral sequences

Hicks Seminar Room J11

Abstract:
I'll discuss recent joint work with Muriel Livernet. We consider the homotopy theory of the category of spectral sequences with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.

Dec 1

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

16:00

The complete graph operad

Hicks Seminar Room J11

Abstract:
The complete graph operad is an E_n-operad, completely combinatorial in nature, and apparently occupying a central position in the world of E_n-operads. This in spite of the fact that up to now there seems to be no (correct) proof in the literature that this operad actually is E_n. I'll discuss some aspects of this operad that I didn't get to in my crash course last spring, but I will try to make the talk independent of what was discussed in that course.

Dec 8

Thu

James Brotherston (Sheffield)

Topology Seminar

16:00

Monoidal model categories relating to spectral sequences

Hicks Seminar Room J11

Abstract:
I'll introduce some model categories of Cirici, Egas Santander, Livernet and Whitehouse on the categories of filtered chain complexes and bicomplexes (as well as some newer intermediary ones indexed by finite non-empty subsets $S$ of the naturals). Their weak equivalences are determined as isomorphisms on the $(r+1)$-page of the associated spectral sequences where $r = \max S$. I'll show that these are all Quillen equivalent via a zig-zag of totalisation and shift-décalage adjunctions so they all present the same homotopy category. I'll also demonstrate the model structures of filtered chains are in fact monoidal model categories satisfying the monoid axiom. By a result of Shipley and Schwede, we then obtain model structures of filtered differential graded algebras with the same weak equivalences enhancing previous work of Halperin and Tanré.

Mar 2

Thu

Ieke Moerdijk (Sheffield)

Topology Seminar

16:15

An elementary approach to bar-cobar duality for functors

Hicks Seminar Room J11

Abstract:
I will explain a version of bar-cobar (or "Koszul") duality between covariant and contravariant functors on a category of trees, the proof of which is elementary and explicit. The (known) duality for linear operads is a special case, as is the (new) extension to linear infinity-operads. Reference: Hoffbeck-Moerdijk, Homology of infinity-operads, Arxiv

Mar 9

Thu

Julie Rasmusen (Warwick)

Topology Seminar

16:00

THR of Poincaré infinity-categories

Hicks Seminar Room J11

Abstract:
In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable infinity-categories. I will introduce the basic ideas and notions of this new theory, but as it is often the case when working with K-theory in any form, this can be very hard to describe. I will therefore introduce a tool which might make our life a bit easier: Real Topological Hochschild Homology. I will explain the ingredients that goes into constructing in particular the geometric fixed points of this as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh.

Mar 30

Thu

Neil Strickland (Sheffield)

Topology Seminar

16:00

Global rational representation theory (joint with Luca Pol)

Hicks Seminar Room J11

Abstract:
Let U be the category of finite groups and conjugacy classes of surjective homomorphisms, or some reasonable subcategory of that. Let A be the category of contravariant functors from U to rational vector spaces (which is equivalent to a certain category of globally equivariant spectra with rational homotopy groups). The category A has some unusual properties: there is a good theory of duality but finitely generated projective objects are not strongly dualisable, all projective objects are injective but not vice-versa, and so on. This makes it difficult to analyse the Balmer spectrum of the associated derived category, but we will explain some progress towards that goal.

Apr 26

Wed

Nick Kuhn (Virginia)

Topology Seminar

16:00

Chromatic Smith Fixed Point Theory

Hicks Seminar Room J11

Abstract:
The study of the action of a finite p-group G on a finite G-CW complex X is one of the oldest topics in algebraic topology. In the late 1930's, P. A. Smith proved that if X is mod p acyclic, then so is X^G, its subspace of fixed points. A related theorem of Ed Floyd from the early 1950's says that the dimension of the mod p homology of X will bound the dimension of the mod p homology of X^G. The study of thick tensored categories in the category of G-spectra has led to the problem of identifying "chromatic" variants of these theorem, with mod p homology replaced by the Morava K-theories (at the prime p). An example of a new chromatic Floyd theorem is the following: if G is a cyclic p-group, then the dimension over K(n)* of K(n)*(X) will bound the dimension over K(n-1)* of K(n-1)*(X^G).
These chromatic fixed point theorems open the door for new applications. For example, one can deduce that a C_2 action on the 5 dimensional Wu manifold will have fixed points that have the rational homology of a sphere. In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava K-theory of some real Grassmanians: this is a non-equivariant result.
An early result in this area was by Neil Strickland. My own contributions have included joint work with Chris Lloyd and also William Balderrama.
In my talk, I'll try to give an overview of some of this.

Apr 27

Thu

Nicola Gambino (Manchester)

Topology Seminar

16:00

The effective model structure

Hicks Seminar Room J11

Abstract:
For a category E with finite limits and well-behaved countable coproducts, we construct a new Quillen model structure on the category of simplicial objects in E, which we call the effective model structure. The effective model structure generalises the Kan-Quillen model structure on simplicial sets; in particular, its fibrant objects can be viewed as infinity-groupoids (i.e. Kan complexes) in E. After introducing the main definitions and outlining the key steps of the proof of the existence of the effective model structure, I will describe some of its peculiar properties and what they mean in terms of its associated infinity-category. This is based on joint work with Simon Henry, Christian Sattler and Karol Szumiło (https://doi.org/10.1017/fms.2022.13).

May 4

Thu

John Greenlees (Warwick)

Topology Seminar

16:00

Rational equivariant cohomology theories for compact Lie groups

Hicks Seminar Room J11

Abstract:
The overall project is to build an algebraic model for rational G-equivariant cohomology theories for all compact Lie groups G. When G is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of G. The talk will focus on structural features of the expected model for general G such as those above, and feature recent joint work with Balchin and Barthel.

May 11

Thu

Luciana Bonatto (MPIM Bonn)

Topology Seminar

16:00

Generalised Configuration Spaces

Hicks Seminar Room J11

Abstract:
Configuration spaces are, on the one hand, powerful invariants and, on the other, spaces with many computable properties. They have also been shown to provide concrete models for homotopy-theoretical constructions such as the free E_n-algebras and the infinite loop spaces associated to stable homotopy theory. These spaces have been generalised in (at least) two directions: the first allows for controlled interactions between the particles of the configuration (for instance allowing some collisions), and the other looks at configurations not of points, but of more general submanifolds. In this talk we will discuss these generalisations, and how they lead to powerful constructions such as factorization homology. We will also discuss in which cases these spaces still carry desirable computational properties seen in the classical configuration spaces, such as homological stability.

May 18

Thu

James Cranch (Sheffield)

Topology Seminar

16:00

What is a polynomial?

Hicks Seminar Room J11

Abstract:
In this mostly expository talk. I'll explain some (different) recipes for defining concepts of "polynomial map" and "polynomial functor" in various settings. I'll explain what some of this has to do with algebraic K-theory, and mention several things I don't know.

May 22

Mon

Jelena Grbic (Southampton)

Topology Seminar

16:00

Higher Whitehead maps in polyhedral products

Hicks Seminar Room J11

Abstract:
We define generalised higher Whitehead maps in polyhedral products. By investigating the interplay between the homotopy-theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products. This is joint work with George Simmons and Matthew Staniforth.

Oct 12

Thu

Daniel Graves (Leeds)

Topology Seminar

16:00

Homology of generalized rook-Brauer algebras

Hicks Seminar Room J11

Abstract:
I will expand on the slogans I gave in last week's gong show. I'll give definitions of some generalizations of rook-Brauer algebras (and their subalgebras) by introducing equivariance and braiding. I'll discuss how we can identify the homology of some of these algebras with the group homology of braid groups and certain semi-direct product groups. I'll also discuss how we can deduce homological stability results and discuss some ideas for future work.

Oct 19

Thu

Neil Strickland (Sheffield)

Topology Seminar

16:00

Double subdivision of relative categories

Hicks Seminar Room J11

Abstract:
By a relative category we mean a category $\mathcal{C}$ equipped with a class $\text{we}$ of weak equivalences. Given such a thing, one can construct a simplicial set $N\mathcal{C}$, called the relative nerve. (In the case where $\text{we}$ is just the class of identity morphisms, this is just the usual nerve of $\mathcal{C}$.) Under mild conditions on $\mathcal{C}$, one can show that $N\mathcal{C}$ is a quasicategory (as defined by Joyal and studied by Lurie), and that the homotopy category of $N\mathcal{C}$ is the category of fractions $\mathcal{C}[\text{we}^{-1}]$. Lennart Meier gave a proof of this, but it depended on quoting a large body of theory related to model categories in the sense of Quillen. I will explain a different approach which instead uses more concrete combinatorial constructions with various specific finite posets.

Oct 26

Thu

Marco Schlichting (Warwick)

Topology Seminar

16:00

On the relation between Hermitian K-theory and Milnor-Witt K-theory

Hicks Seminar Room J11

Abstract:
Hermitian K-theory of a commutative ring R is the algebraic K-theory of finitely generated projective R-modules equipped with a non-degenerate symmetric/symplectic/quadratic form. The algebra generated in degree (1,1) modulo the Steinberg relation in degree (2,2) is called Milnor-Witt K-theory and plays an important role in A1-homotopy theory. Multiplicativity of Hermitian K-theory defines a graded ring homomorphism from Milnor-Witt K-theory to Hermitian K-theory. We prove a homology stability result for symplectic groups and use this to construct a map from Hermitian K-theory of a local ring to Milnor-Witt K-theory in degrees 2,3 mod 4. Finally, we compute the composition of the maps from Milnor-Witt to Hermitian and back to Milnor-Witt K-theory as multiplication with a particular integral form.

Nov 2

Thu

Alex Corner (Sheffield Hallam)

Topology Seminar

16:30

Weak Vertical Composition

Hicks Seminar Room J11

Abstract:
A usual test for a suitable semi-strict notion of n-category is that in its degenerate cases, it produces particular lower-dimensional monoidal structures as predicted by Baez and Dolan's Stabilisation Hypothesis. These structures are of interest in topology in that they produce algebraic homotopy n-types which are not equivalent to a fully strict notion of n-category. We are concerned with doubly-degenerate tricategories, which should produce a structure equivalent to a braided monoidal category. Gordon, Power, and Street show that in the case of Gray-categories, where interchange of 2-cells is weak but all other composition is strict, this is certainly the case. Joyal and Kock show further that the weakness, like a bump under a carpet, can be pushed solely into the horizontal units for 2-cells, and that this notion also matches braided monoidal categories in the doubly-degenerate case. In this talk I will introduce a notion of tricategory in which only the vertical composition of 2-cells is weak. These will be identified with categories strictly enriched in the category of bicategories and strict 2-functors with cartesian monoidal product, which, although constituting an unusual mix of weakness and strictness allows a very straightforward algebraic characterisation of weak vertical tricategories using the theory of 2-monads and 2-distributive laws. Thus far only object-level correspondences have been considered, but we show that with special consideration given to icon-like higher cells, we can form a 2-categorical totality of these degenerate structures, along with their weak maps and transformations, allowing us to give a full comparison with the 2-category of braided monoidal categories.

Nov 16

Thu

Callum Reader (Sheffield)

Topology Seminar

16:00

Optimal Transport from Enriched Categories

Hicks Seminar Room J11

Abstract:
Imagine we have a metric space whose points we think of as warehouses, and whose distances give the cost of moving a unit of stock. Now imagine we have two probability distributions that tell us how much stock is in each warehouse. A classical problem from optimal transport theory asks: how we might rearrange one distribution into another with minimal cost? The 'minimal cost' in this scenario defines a metric on the space of all probability measures, this metric is called earth-mover's distance. Now instead of a metric space imagine we have a category enriched over the extended non-negative reals. As Lawvere points out, these enriched categories can be thought of as generalised metric spaces. We show that from this perspective, probability measures might be thought of as functors and the natural transformation object that exists between them is actually equal to the earth-mover's distance. What's more, we show that, when we take consider sub-probability measures - that is, measures with total mass less than one - the natural transformation object improves on the earth-mover's distance and can be intuited as the 'minimal cost of meeting demand'.

Nov 23

Thu

Yuqing Shi (MPIM Bonn)

Topology Seminar

16:00

Costabilisation of telescopic spectral Lie algebras

Hicks Seminar Room J11

Abstract:
One can think of the stabilisation of an ∞-category as the ∞-category of objects that admit infinite deloopings. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that allow infinite desuspensions. It is easy to see that the costabilisation of the ∞-category of homotopy types is trivial. Fix a prime number p. In this talk I will show that the costablisation of the ∞-category of T(h)-local spectral Lie algebras is equivalent to the ∞-category of T(h)-local spectra, where T(h) denotes a p-local telescope spectrum of height h. A key ingredient of the proof is to relate spectral Lie algebras to (spectral) Eₙ algebras via Koszul duality.

Nov 30

Thu

Fiona Torzewska (Bristol)

Topology Seminar

16:00

Motion groupoids

Hicks Seminar Room J11

Abstract:
The braiding statistics of point particles in 2-dimensional topological phases are given by representations of the braid groups. One approach to the study of generalised particles in topological phases, loop particles in 3-dimensions for example, is to generalise (some of) the several different realisations of the braid group. In this talk I will construct for each manifold M its motion groupoid $Mot_M$, whose object class is the power set of M. I will discuss several different, but equivalent, quotients on motions leading to the motion groupoid. In particular that the quotient used in the construction $Mot_M$ can be formulated entirely in terms of a level preserving isotopy relation on the trajectories of objects under flows -- worldlines (e.g. monotonic `tangles'). I will also give a construction of a mapping class groupoid $\mathrm{MCG}_M$ associated to a manifold M with the same object class. For each manifold M I will construct a functor $F \colon Mot_M \to MCG_M$, and prove that this is an isomorphism if $\pi_0$ and $\pi_1$ of the appropriate space of self-homeomorphisms of M is trivial. In particular there is an isomorphism in the physically important case $M=[0,1]^n$ with fixed boundary, for any $n\in\mathbb{N}$. I will discuss several examples throughout.

Dec 7

Thu

Lukas Brantner (Oxford)

Topology Seminar

16:00

Deformations and lifts of Calabi-Yau varieties in characteristic p

Hicks Seminar Room J11

Abstract:
Homotopy theory allows us to study infinitesimal deformations of algebraic varieties via (partition) Lie algebras. We apply this general principle to two classical problems on Calabi-Yau varieties Z in characteristic p. First, we show that if Z has torsion-free crystalline cohomology and degenerating Hodge-de Rham spectral sequence, then its mixed characteristic deformations are unobstructed. This generalises the BTT theorem to characteristic p. If Z is ordinary, we show that it moreover admits a canonical (and algebraisable) lift to characteristic zero, thereby extending Serre-Tate theory to Calabi-Yau varieties. This is joint work with Taelman, and generalises results of Achinger-Zdanowicz, Bogomolov-Tian- Todorov, Deligne-Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre-Tate, and Ward.

Dec 14

Thu

Simon Willerton (Sheffield)

Topology Seminar

16:00

Parametrized mates, or how I finally understood Fausk, Hu and May.

Hicks Seminar Room J11

Abstract:
In various parts of mathematics such as algebraic geometry, homotopy theory and representation theory, you can encounter situations where you have a strong monoidal functor $f^*$ with an adjoint $f_+$. One automatically gets a comparison map between $f_+(a \times f^*b)$ and $f_+(a) \times b$ where $\times$ is the monoidal product. The projection formula is said to hold when this comparison map is an isomorphism. Fausk, Hu and May showed that the projection formula holds under various conditions, such as $f^*$ being a strong closed monoidal functor. I will show how a theory of mates for parametrized adjunctions (and my graphical version of it) has helped me understand their work.

Feb 8

Thu

Sarah Whitehouse (Sheffield)

Topology Seminar

16:00

Homotopy theory of spectral sequences

Hicks Seminar Room J11

Abstract:
For each r, maps which are quasi-isomorphisms on the r page provide a class of weak equivalences on the category of spectral sequences. The talk will cover homotopy theory associated with this setting. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures. One of these has the property that spectral sequences is a homotopically full subcategory and so, by results of Meier, exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories. We also note how the presheaf approach provides some insight into the décalage functor on spectral sequences. This is joint work with Muriel Livernet.

Feb 22

Thu

Joseph Grant

Topology Seminar

16:00

Frobenius algebra objects in Temperley-Lieb categories at roots of unity

Hicks Seminar Room J11

Abstract:
Frobenius algebras appear in many parts of maths and have nice properties. One can define algebra objects in any monoidal category, and there is a standard definition of when such an algebra object is Frobenius. But this definition is not satisfied by something which we'd like to think of as an algebra object in Temperley-Lieb categories at roots of unity. We will explore a more general definition of a Frobenius algebra object which covers this example, and will explore some of its properties. This is joint work with Mathew Pugh.

Feb 29

Thu

Jack Romo (Leeds)

Topology Seminar

16:00

$(\infty, 2)$-Categories and their Homotopy Bicategories

Hicks Seminar Room J11

Abstract:
Across the multitude of definitions for a higher category, a dividing line can be found between two major camps of model. On one side lives the ‘algebraic’ models where composition operations between morphisms are given, like Bénabou’s bicategories, tricategories following Gurski and the models of n-category of Batanin and Leinster, Trimble and Penon. On the other end, one finds the ‘non-algebraic’ models, where the space of possible composites is only guaranteed to be contractible. These include the models of Tamsamani and Paoli, along with quasicategories, Segal n-categories, complete n-fold Segal spaces and more. The bridges between these models remain somewhat mysterious. Progress has been made in certain instances, as seen in the work of Tamsamani, Leinster, Lack and Paoli, Cottrell, Campbell, Nikolaus and others. Nonetheless, the correspondence remains incomplete; indeed, for instance, there is no fully verified means in the literature to take an `algebraic’ homotopy n-category of any known model of $(\infty, n)$-category for general n. In this talk, I will present my contributions to the problem of taking algebraic homotopy bicategories of non-algebraic $(\infty, 2)$-categories. This talk also serves as an introduction to the model of $(\infty, 2)$-category I will be using, namely complete 2-fold Segal spaces. If time permits, I will discuss how to compute the fundamental bigroupoid of a topological space with this construction.

Mar 7

Thu

Nadia Mazza (Lancaster)

Topology Seminar

16:00

Endotrivial modules for finite groups of Lie type

Hicks Seminar Room J11

Abstract:
Let G be a finite group and k a field of positive characteristic p diving the order of G. An endotrivial kG-module is a finitely generated kG-module which is "invertible" in some suitable sense. Since the late 70s, these modules have been intensely studied in modular representation theory. In this talk, we review the essential background on endotrivial modules, and present some results about endotrivial modules for finite groups of Lie type, obtained jointly with Carlson, Grodal and Nakano.

Mar 21

Thu

Andy Baker (Glasgow)

Topology Seminar

16:00

Endotrivial modules for the quaternion group and iterated Jokers in chromatic homotopy theory

Hicks Seminar Room J11

Abstract:
The Joker is a famous very singular example of an endotrivial module over the 8-dimension subHopf algebra of the mod 2 Steenrod algebra generated by $\operatorname{Sq}^1$ and $\operatorname{Sq}^2$. It is known that this can be realised as the cohomology of two distinct Spanier-Whitehead dual spectra. Similarly, the double and iterated double are also realisable, but then the process stops.

Apr 18

Thu

Briony Eldridge (Southampton)

Topology Seminar

16:00

Loop Spaces of Polyhedral Products Associated with Substitution Complexes

Hicks Seminar Room J11

Abstract:
Polyhedral products are a topological space formed by gluing together ingredient spaces in a manner governed by a simplicial complex. They appear in many areas of study, including toric topology, combinatorics, commutative algebra, complex geometry and geometric group theory. A fundamental problem is to determine how operations on simplicial complexes change the topology of the polyhedral product. In this talk, we consider the substitution complex operation. We obtain a description of the loop space associated with some substitution complexes, and use this to build a new family of simplicial complexes such that the homotopy type of the loop space of the moment angle complex is a product of spheres and loops on spheres.

May 2

Thu

Ehud Meir (Aberdeen)

Topology Seminar

16:00

Invariants that are covering spaces and their Hopf algebras

Hicks Seminar Room J11

Abstract:
Different flavours of string diagrams arise naturally in studying algebraic structures (e.g. algebras, Hopf algebras, Frobenius algebras) in monoidal categories. In particular, closed diagrams can be realized as scalar invariants. For a structure of a given type the closed diagrams form a commutative algebra that has a richer structure of a self dual Hopf algebra. This is very similar, but not quite the same, as the positive self adjoint Hopf algebras that were introduced by Zelevinsky in studying families of representations of finite groups. In this talk I will show that the algebras of invariants admit a lattice that is a PSH-algebra. This will be done by considering maps between invariants, and realizing them as covering spaces. I will then show some applications to subgroup growth questions, and a formula that relates the Kronecker coefficients to finite index subgroups of free groups.

May 9

Thu

Georg Struth (Sheffield)

Topology Seminar

16:00

Single-set Cubical Categories and Their Formalisation with a Proof Assistant

Hicks Seminar Room J11

Abstract:
Cubical sets and cubical categories are widely used in mathematics and computer science, from homotopy theory to homotopy type theory, higher-dimensional automata and, last but not least, higher-dimensional rewriting, where our own interest in these structures lies. To formalise cubical categories with the Isabelle/HOL proof assistant along the path of least resistance, we take a single-set approach to categories, which leads to new axioms for cubical categories. Taming the large number of initial candidate axioms has relied essentially on Isabelle's proof automation. Yet we justify their correctness relative to the standard axiomatisation by Al Agl, Brown and Steiner via categorical equivalence proofs outside of Isabelle. In combination, these results present a case study in experimental mathematics with a proof assistant. In this talk I will focus on the formalisation experience -- lights and shadows -- and conclude with some general remarks about formalised mathematics. This is joint work with Philippe Malbos and Tanguy Massacrier (Université Claude Bernard Lyon 1).

May 16

Thu

Gong Show

Topology Seminar

15:30

Hicks Seminar Room J11

May 16

Thu

Gong Show

Topology Seminar

16:30

Hicks Seminar Room J11