Forthcoming events in this series


Mon, 26 Feb 2024
15:30
L4

Morava K-theory of infinite groups and Euler characteristic

Irakli Patchkoria
(University of Aberdeen)
Abstract

Given an infinite discrete group G with a finite model for the classifying space for proper actions, one can define the Euler characteristic of G and the orbifold Euler characteristic of G. In this talk we will discuss higher chromatic analogues of these invariants in the sense of stable homotopy theory. We will study the Morava K-theory of G and associated Euler characteristic, and give a character formula for the Lubin-Tate theory of G. This will generalise the results of Hopkins-Kuhn-Ravenel from finite to infinite groups and the K-theoretic results of Adem, Lück and Oliver from chromatic level one to higher chromatic levels. At the end we will mention explicit computations for some arithmetic groups and mapping class groups in terms of class numbers and special values of zeta functions. This is all joint with Wolfgang Lück and Stefan Schwede.

Mon, 19 Feb 2024
15:30
L4

Maps between spherical group algebras

Thomas Nikolaus
(Universitaet Muenster)
Abstract

I will speak about a central question in higher algebra (aka brave new algebra), namely which rings or schemes admit 'higher models', that is lifts to the sphere spectrum. This question is in some sense very classical, but there are many open questions. These questions are closely related to questions about higher versions of prismatic cohomology and delta ring, asked e.g. by Scholze and Lurie. Concretely we will consider the case of group algebras and explain how to understand maps between lifts of group algebras to the sphere spectrum. The results we present are joint with Carmeli and Yuan and on the prismatic side with Antieau and Krause.

Mon, 12 Feb 2024
15:30
L4

A filtration of handlebody Teichmüller space

Ric Wade
((Oxford University))
Abstract

The handlebody group is defined to be the mapping class group of a handelbody (rel. boundary). It is a subgroup of the mapping class group of the surface of the handlebody, and maps onto the outer automorphism group of its fundamental group (the free group of rank equal to its genus). 

Recently Hainaut and Petersen described a subspace of moduli space forming an orbifold classifying space for the handlebody group, and combined this with work of Chan-Galatius-Payne to construct cohomology classes in the group. I will talk about how one can build on their ideas to define a cocompact EG for the handlebody group inside Teichmüller space. This is a manifold with boundary and comes with a filtration by labelled disk systems which we call the `RGB (red-green-blue) disk complex.' I will describe this filtration, use it to describe the boundary of the manifold, and speculate about potential applications to duality results. Based on work-in-progress with Dan Petersen.

Mon, 05 Feb 2024
15:30
L4

Bicommutant categories

Andre Henriques
((Oxford University))
Abstract

Bicommutant categories, initially invented for the purposes of Chern-Simons theory and 2d CFT, seem to also appear in other domains of math with examples related to group theory, and dynamical systems.

Mon, 29 Jan 2024
15:30
L4

Categorifying the four color theorem with applications to Gromov-Witten theory

Scott Baldridge
(Louisiana State University)
Abstract
The four color theorem states that each bridgeless planar graph has a proper $4$-face coloring. It can be generalized to certain types of CW complexes of any closed surface for any number of colors, i.e., one looks for a coloring of the 2-cells (faces) of the complex with $m$ colors so that whenever two 2-cells are adjacent to a 1-cell (edge), they are labeled different colors.

In this talk, I show how to categorify the $m$-color polynomial of a surface with a CW complex. This polynomial is based upon Roger Penrose’s seminal 1971 paper on abstract tensor systems and can be thought of as the ``Jones polynomial’’ for CW complexes. The homology theory that results from this categorification is called the bigraded $m$-color homology and is based upon a topological quantum field theory (that will be suppressed from this talk due to time). The construction of this homology shares some similar features to the construction of Khovanov homology—it has a hypercube of states, multiplication and comultiplication maps, etc. Most importantly, the homology is the $E_1$ page of a spectral sequence whose $E_\infty$ page has a basis that can be identified with proper $m$-face colorings, that is, each successive page of the sequence provides better approximations of $m$-face colorings than the last. Since it can be shown that the $E_1$ page is never zero, it is safe to say that a non-computer-based proof of the four color theorem resides in studying this spectral sequence! (This is joint work with Ben McCarty.)

If time, I will relate this work to the study of the moduli space of stable genus $g$ curves with $n$ marked points. Using Strebel quadratic differentials, one can identify this moduli space with a subspace of the space of metric ribbon graphs with labeled boundary components. Proper $m$-face coloring in this setup is, in a sense, studying points in the space of metric ribbon graphs where similarly-colored boundaries (marked points) don’t get ``too close’’ to each other. We will end with some speculations about what this might mean for Gromov-Witten theory of Calabi-Yau manifolds.
 
Note to students: This talk will be hands-on with ideas explained through the calculation of examples. Graduate students and researchers who are interested in graph theory, topology, or representation theory are encouraged to attend.   
 
Mon, 22 Jan 2024
15:30

Surface automorphisms and elementary number theory

Greg McShane
(Universite Grenoble-Alpes)
Abstract
The modular surface $\mathbb{H}/\Gamma,\, \Gamma= \mathrm{SL}(2,\mathbb{Z})$ has many covers - for example the three punctured torus $\mathbb{H}/\Gamma(2)$ and the once punctured torus $\mathbb{H}/\Gamma'$. We will discuss how classical Diophantine approximation can be interpreted in terms of the behaviour of geodesics on the once punctured torus and a geometric reformulation of the Frobenius uniqueness conjecture.
We will then give an account of two theorems of Fermat in terms of   the automorphisms of $\mathbb{H}/\Gamma(2)$:
- if $p$ is a prime such that $4|(p-1)$ then  can be written as a   sum of squares $p = c^2 + d^2$
- if $p$ is a prime such that $3|(p-1)$ then  can be written as  $  p = c^2 +cd +  d^2$
Finally we will discuss possible extensions to surfaces of the for  m $\mathbb{H}/\Gamma_0(N)$.
 
Mon, 15 Jan 2024
15:30

Invariant splittings of HFK of satellite knots

Sungkyung Kang
((Oxford University))
Abstract

Involutive knot Floer homology, a refinement of knot Floer theory, is a powerful knot invariant which was used to solve several long-standing problems, including the one-is-not-enough result for 4-manifolds with boundary. In this talk, we show that if the involutive knot Floer homology of a knot K admits an invariant splitting, then the induced splitting if the knot Floer homology of P(K), for any pattern P, can be made invariant under its \iota_K involution. As an application, we construct an infinite family of examples of pairs of exotic contractible 4-manifolds which survive one stabilization, and observe that some of them are potential candidates for surviving two stabilizations.
 

Mon, 27 Nov 2023
15:30
L4

Costabilisation of telescopic spectral Lie algebras

Yuqing Shi
(Max Planck Institute for Mathematics)
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.
 

Mon, 20 Nov 2023
15:30
L4

Quantum field theory of Lorentzian manifolds

Alexander Schenkel
(University of Nottingham)
Abstract

In this talk I will provide an overview of our current research at the interface of quantum field theory (QFT), Lorentzian geometry and higher categorical structures. I will present operads which encode the rich algebraic structure of QFTs on Lorentzian manifolds and show that in low dimensions their algebras relate to familiar algebraic structures. Our operads share certain similarities with the little disk operads from topology, in particular they involve a homotopical localization at geometric embeddings related to ‘time evolution’. I will show that, in contrast to the topological context, this homotopical localization can be strictified in many important classes of examples, which is loosely speaking due to the 1-dimensional nature of time evolution in Lorentzian geometry. I will conclude by explaining how simple examples of such Lorentzian QFTs can be constructed from a homotopical generalization of the concept of Green’s operators for hyperbolic partial differential equations, which we call Green hyperbolic complexes. Throughout this talk, I will frequently comment on the similarities and differences between our approach, factorization algebras and functorial field theories.

Mon, 06 Nov 2023
15:30
L4

Understanding infinite groups via their actions on Banach spaces

Cornelia Drutu
((Oxford University) )
Abstract

One way of studying infinite groups is by analysing
 their actions on classes of interesting spaces. This is the case
 for Kazhdan's property (T) and for Haagerup's property (also called a-T-menability),
 formulated in terms of actions on Hilbert spaces and relevant in many areas
(e.g. for the Baum-Connes conjectures, in combinatorics, for the study of expander graphs, in ergodic theory, etc.)
 
Recently, these properties have been reformulated for actions on Banach spaces,
with very interesting results. This talk will overview some of these reformulations
 and their applications. Part of the talk is on joint work with Ashot Minasyan and Mikael de la Salle, and with John Mackay.
 

Mon, 30 Oct 2023
15:30
L4

Quantitative implications of positive scalar curvature.

Thomas RICHARD
(Université Paris Est Créteil)
Abstract

Until the 2010’s the only « comparison geometry » result for compact Riemannian manifolds (M^n,g) with scal≥n(n-1) was Greene’s upper bound on the injectivity radius. Moreover, it is known that classical metric invariants (volume, diameter) cannot be controlled by a lower bound on the scalar curvature alone. It has only recently been discovered that some more subtle invariants, such as 2-systoles, can be controlled under a lower bounds on scal provided M has enough topology. We will present some results of Bray-Brendle-Neves (in dim 3), Zhu (in dim≤7) for S^2xT^(n-2), some version for S^2xS^2 and some conjecture with more general topology which we show to hold true under the additional assumption of Kaehlerness.

Mon, 23 Oct 2023
15:30
L4

Khovanov homology and the Fukaya category of the three-punctured sphere

Claudius Zibrowius
(Durham University)
Abstract

About 20 years ago, Dror Bar-Natan described an elegant generalisation
of Khovanov homology to tangles with any number of endpoints, by
considering certain quotients of two-dimensional relative cobordism
categories.  I claim that these categories are in general not
well-understood (not by me in any case).  However, if we restrict to
tangles with four endpoints, things simplify and Bar-Natan's category
turns out to be closely related to the wrapped Fukaya category of the
four-punctured sphere.  This relationship gives rise to a symplectic
interpretation of Khovanov homology that is useful both for doing
calculations and for proving theorems.  I will discuss joint work in
progress with Artem Kotelskiy and Liam Watson where we investigate what
happens when we fill in one of the punctures.
 

Mon, 16 Oct 2023
15:30
L4

Algorithms for Seifert fibered spaces

Adele Jackson
((Oxford University))
Abstract

Given two mathematical objects, the most basic question is whether they are the same. We will discuss this question for triangulations of three-manifolds. In practice there is fast software to answer this question and theoretically the problem is known to be decidable. However, our understanding is limited and known theoretical algorithms could have extremely long run-times. I will describe a programme to show that the 3-manifold homeomorphism problem is in the complexity class NP, and discuss the important sub-case of Seifert fibered spaces. 

 

Mon, 09 Oct 2023
15:30
L4

Distribution of minimal surfaces in compact hyperbolic 3-manifolds

Ilia Smilga
((Oxford University))
Abstract

In a classical work, Bowen and Margulis proved the equidistribution of
closed geodesics in any hyperbolic manifold. Together with Jeremy Kahn
and Vladimir Marković, we asked ourselves what happens in a
three-manifold if we replace curves by surfaces. The natural analog of a
closed geodesic is then a minimal surface, as totally geodesic surfaces
exist only very rarely. Nevertheless, it still makes sense (for various
reasons, in particular to ensure uniqueness of the minimal
representative) to restrict our attention to surfaces that are almost
totally geodesic.

The statistics of these surfaces then depend very strongly on how we
order them: by genus, or by area. If we focus on surfaces whose *area*
tends to infinity, we conjecture that they do indeed equidistribute; we
proved a partial result in this direction. If, however, we focus on
surfaces whose *genus* tends to infinity, the situation is completely
opposite: we proved that they then accumulate onto the totally geodesic
surfaces of the manifold (if there are any).

Mon, 12 Jun 2023
15:30
L5

On the Dualizability of Fusion 2-Categories

Thibault Decoppet
Abstract

Fusion 2-categories were introduced by Douglas and Reutter so as to define a state-sum invariant of 4-manifolds. Categorifying a result of Douglas, Schommer-Pries and Snyder, it was conjectured that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object in an appropriate symmetric monoidal 4-category. I will sketch a proof of this conjecture, which will proceed by studying, and in fact classifying, the Morita equivalence classes of fusion 2-categories. In particular, by appealing to the cobordism hypothesis, we find that every fusion 2-category yields a fully extended framed 4D TQFT. I will explain how these theories are related to the ones constructed using braided fusion 1-categories by Brochier, Jordan, and Snyder.

Mon, 22 May 2023
15:30
L5

Combining the minimal-separating-set trick with simplicial volume

Hannah Alpert
Abstract

In 1983 Gromov proved the systolic inequality: if M is a closed, essential n-dimensional Riemannian manifold where every loop of length 2 is null-homotopic, then the volume of M is at least a constant depending only on n.  He also proved a version that depends on the simplicial volume of M, a topological invariant generalizing the hyperbolic volume of a closed hyperbolic manifold.  If the simplicial volume is large, then the lower bound on volume becomes proportional to the simplicial volume divided by the n-th power of its logarithm.  Nabutovsky showed in 2019 that Papasoglu's method of area-minimizing separating sets recovers the systolic inequality and improves its dependence on n.  We introduce simplicial volume to the proof, recovering the statement that the volume is at least proportional to the square root of the simplicial volume.

Mon, 15 May 2023
15:30
L5

Virtual classes of character stacks

Marton Hablicsek
Abstract

Questions about the geometry of G-representation varieties on a manifold M have attracted many researchers as the theory combines the algebraic geometry of G, the topology of M, and the group theory and representation theory of G and the fundamental group of M. In this talk, I will explain how to construct a Topological Quantum Field Theory to compute virtual classes of character stacks (G-representation varieties equipped with the adjoint G-action) in the Grothendieck ring of stacks. I will also show a few features of the construction (for instance, how to obtain arithmetic information) focusing on a couple of simple examples.
The work is joint with Jesse Vogel and Ángel González-Prieto.  

Mon, 15 May 2023
14:00
C6

Ext in functor categories and stable cohomology of Aut(F_n) (Arone)

Greg Arone
Abstract

 

We present a homotopy-theoretic method for calculating Ext groups between polynomial functors from the category of (finitely generated, free) groups to abelian groups. It enables us to substantially extend the range of what can be calculated. In particular, we can calculate torsion in the Ext groups, about which very little has been known. We will discuss some applications to the stable cohomology of Aut(F_n), based on a theorem of Djament.   

 

 

Mon, 24 Apr 2023
15:30
L5

Coarse embeddings, and yet more ways to avoid them

David Hume
(Bristol)
Abstract

Coarse embeddings (maps between metric spaces whose distortion can be controlled by some function) occur naturally in various areas of pure mathematics, most notably in topology and algebra. It may therefore come as a surprise to discover that it is not known whether there is a coarse embedding of three-dimensional real hyperbolic space into the direct product of a real hyperbolic plane and a 3-regular tree. One reason for this is that there are very few invariants which behave monotonically with respect to coarse embeddings, and thus could be used to obstruct coarse embeddings.


 

Mon, 06 Mar 2023
15:30
L4

Homeomorphisms of surfaces: a new approach

Richard Webb
(University of Manchester)
Abstract

Despite their straightforward definition, the homeomorphism groups of surfaces are far from straightforward. Basic algebraic and dynamical problems are wide open for these groups, which is a far cry from the closely related and much better understood mapping class groups of surfaces. With Jonathan Bowden and Sebastian Hensel, we introduced the fine curve graph as a tool to study homeomorphism groups. Like its mapping class group counterpart, it is Gromov hyperbolic, and can shed light on algebraic properties such as scl, via geometric group theoretic techniques. This brings us to the enticing question of how much of Thurston's theory (e.g. Nielsen--Thurston classification, invariant foliations, etc.) for mapping class groups carries over to the homeomorphism groups. We will describe new phenomena which are not encountered in the mapping class group setting, and meet some new connections with topological dynamics, which is joint work with Bowden, Hensel, Kathryn Mann and Emmanuel Militon. I will survey what's known, describe some of the new and interesting problems that arise with this theory, and give an idea of what's next.

 

Mon, 27 Feb 2023
15:30
L4

SL(2,C)-character varieties of knots and maps of degree 1

Raphael Zentner
(Durham University)
Abstract

We ask to what extend the SL(2,C)-character variety of the
fundamental group of the complement of a knot in S^3 determines the
knot. Our methods use results from group theory, classical 3-manifold
topology, but also geometric input in two ways: the geometrisation
theorem for 3-manifolds, and instanton gauge theory. In particular this
is connected to SU(2)-character varieties of two-component links, a
topic where much less is known than in the case of knots. This is joint
work with Michel Boileau, Teruaki Kitano, and Steven Sivek.

Mon, 20 Feb 2023
15:45

Factorization homology of braided tensor categories

Adrien Brochier
(Paris)
Abstract

Factorization homology is an arguably abstract formalism which produces
well-behaved topological invariants out of certain "higher algebraic"
structures. In this talk, I'll explain how this formalism can be made
fairly concrete in the case where this input algebraic structure is a
braided tensor category. If the category at hand is semi-simple, this in
fact essentially recovers skein categories and skein algebras. I'll
present various applications of this formalism to quantum topology and
representation theory.
 

Mon, 13 Feb 2023
15:30
Online

Classifying sufficiently connected manifolds with positive scalar curvature

Yevgeny Liokumovich
(University of Toronto)

Note: we would recommend to join the meeting using the Teams client for best user experience.

Abstract

I will describe the proof of the following classification result for manifolds with positive scalar curvature. Let M be a closed manifold of dimension $n=4$ or $5$ that is "sufficiently connected", i.e. its second fundamental group is trivial (if $n=4$) or second and third fundamental groups are trivial (if $n=5$). Then a finite covering of $M$ is homotopy equivalent to a sphere or a connect sum of $S^{n-1} \times S^1$. The proof uses techniques from minimal surfaces, metric geometry, geometric group theory. This is a joint work with Otis Chodosh and Chao Li.
 

Mon, 06 Feb 2023
15:30
L4

The infinitesimal tangle hypothesis

Joost Nuiten (Toulouse)
Abstract

The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension d can be organized into an E_d-monoidal n-category, where n is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free E_d-monoidal n-category with duals generated by a single object.

In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tangle hypothesis: Instead of showing that the E_d-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides supporting evidence for the tangle hypothesis, but can also be used to reduce the tangle hypothesis to a statement at the level of E_d-monoidal (n+1, n)-categories by means of obstruction theory.