15:30

### Coxeter groups acting on CAT(0) cube complexes

## Abstract

We will give a general overview of how one gets groups to act on CAT(0) cube complexes, how compatible such actions are and how this plays out in the setting of Coxeter groups.

Forthcoming events in this series

Mon, 20 Jun 2022

15:30

15:30

L5

Michah Sageev

We will give a general overview of how one gets groups to act on CAT(0) cube complexes, how compatible such actions are and how this plays out in the setting of Coxeter groups.

Mon, 06 Jun 2022

15:30 -
16:30

L5

Ian Agol

(Berkeley)

We show that ribbon concordance forms a partial ordering on the set of knots, answering a question of Gordon. The proof makes use of representation varieties of the knot groups to S O(N) and relations between them induced by a ribbon concordance.

Mon, 30 May 2022

15:30 -
16:30

L5

Severin Bunk

(Oxford)

Gerbes are geometric objects describing the third integer cohomology group of a manifold and the B-field in string theory. Like line bundles, they admit connections and gauge symmetries. In contrast to line bundles, however, there are now isomorphisms between gauge symmetries: the gauge group of a gerbe is a smooth 2-group. Starting from a hands-on example, I will explain gerbes and some of their properties. The main topic of this talk will then be the study of symmetries of gerbes on a manifold with G-action, and how these symmetries assemble into smooth 2-group extensions of G. In the last part, I will survey how this construction can be used to provide a new smooth model for the String group, via a theory of ∞-categorical principal bundles and group extensions.

Mon, 23 May 2022

15:30 -
16:30

L5

Alice Kerr

(Oxford)

A standard question in group theory is to ask if we can categorise the subgroups of a group in terms of their growth. In this talk we will be asking this question for uniform product set growth, a property that is stronger than the more widely understood notion of uniform exponential growth. We will see how considering acylindrical actions on hyperbolic spaces can help us, and give a particular application to mapping class groups.

Mon, 16 May 2022

15:30 -
16:30

L5

Ric Wade

(Oxford)

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a Duality Group, where the isomorphism between homology and cohomology can be twisted by what they called a Dualizing Module. Examples of these groups in geometric group theory (after passing to a finite-index subgroup) include $GL(n,\mathbb{Z})$, mapping class groups, and automorphism groups of free groups.

In work-in-progress with Thomas Wasserman we are looking into the following puzzle: the examples of duality groups that we know of that do not come from manifolds all have classifying spaces that satisfy a weaker local condition called the Cohen-Macaulay property. These spaces also satisfy weaker (twisted) versions of Poincaé duality via their local homology sheaves (or local cohomology cosheaves), and we are attempting to understand more about the links between these geometric versions of duality and the algebraic notion of a duality group. The goal of the talk is to explain more about the words used in the above paragraphs and say where we have got to so far.

Mon, 09 May 2022

15:30 -
16:30

L4

Armando Martino

(Southampton)

We will review some open questions about automorphisms of free groups, give some partial answers, and explain the deformation spaces of trees that they act on, as well as the geometry of these spaces arising from the Lipschitz metric. This will be a gentle introduction to the topic, focused on introducing the concepts.

Mon, 02 May 2022

15:30 -
16:30

Online

Rufus Willett

(University of Hawaii )

Let X be a closed Riemannian manifold, and represent the algebra C(X) of continuous functions on X on the Hilbert space L^2(X) by multiplication. Inspired by the heat kernel proof of the Atiyah-Singer index theorem, I'll explain how to describe K-homology (i.e. the dual theory to Atiyah-Hirzebruch K-theory) in terms of parametrized families of operators on L^2(X) that get more and more 'local' in X as time tends to infinity.

I'll then switch perspectives from C(X) -- the prototypical example of a commutative C*-algebra -- to noncommutative C*-algebras built from discrete groups, and explain how the underlying large-scale geometry of the groups can give rise to approximate 'decompositions' of the C*-algebras. I'll then explain how to use these decompositions and localization in the sense above to compute K-homology, and the connection to some conjectures in topology, geometry, and C*-algebra theory.

Mon, 25 Apr 2022

15:30 -
16:30

L4

Professor Marc Lackenby

((Oxford University) )

Knot theory is divided into several subfields. One of these is hyperbolic knot theory, which is focused on the hyperbolic structure that exists on many knot complements. Another branch of knot theory is concerned with invariants that have connections to 4-manifolds, for example the knot signature and Heegaard Floer homology. In my talk, I will describe a new relationship between these two fields that was discovered with the aid of machine learning. Specifically, we show that the knot signature can be estimated surprisingly accurately in terms of hyperbolic invariants. We introduce a new real-valued invariant called the natural slope of a hyperbolic knot in the 3-sphere, which is defined in terms of its cusp geometry. Our main result is that twice the knot signature and the natural slope differ by at most a constant times the hyperbolic volume divided by the cube of the injectivity radius. This theorem has applications to Dehn surgery and to 4-ball genus. We will also present a refined version of the inequality where the upper bound is a linear function of the volume, and the slope is corrected by terms corresponding to short geodesics that have odd linking number with the knot. My talk will outline the proofs of these results, as well as describing the role that machine learning played in their discovery.

This is joint work with Alex Davies, Andras Juhasz, and Nenad Tomasev

Mon, 07 Mar 2022

15:30

15:30

L5

Joel Hass

(University of California Davis)

Abstract: Almost everything we encounter in our 3-dimensional world is a surface - the outside of a solid object. Comparing the shapes of surfaces is, not surprisingly, a fundamental problem in both theoretical and applied mathematics. Results from the mathematical theory of surfaces are now being used to study objects such as bones, brain cortices, proteins and biomolecules. This talk will discuss recent joint work with Patrice Koehl that introduces a new metric on the space of Riemannian surfaces of genus-zero and some applications to biological surfaces.

Mon, 28 Feb 2022

15:30

15:30

L5

Christoph Weis

Let $G$ be a compact connected Lie group and $k \in H^4(BG,\mathbb{Z})$ a cohomology class. The String 2-group $G_k$ is the central extension of $G$ by the smooth 2-group $BU(1)$ classified by $k$. It has a close relationship to the level $k$ extension of the loop group $LG$.

We will introduce smooth 2-groups and the associated notion of centre. We then compute this centre for the String 2-groups, leveraging the power of maximal tori familiar from classical Lie theory.

The centre turns out to recover the invertible positive energy representations of $LG$ at level $k$ (as long as we exclude factors of $E_8$ at level 2).

Mon, 21 Feb 2022

15:30

15:30

L5

Abigail Thompson

Gay and Kirby formulated a new way to decompose a (closed, orientable) 4-manifold M, called a trisection. I’ll describe how to translate from a classical framed link diagram for M to a trisection diagram. The links so obtained lie on Heegaard surfaces in the 3-sphere, and have surgeries yielding some number of copies of S^1XS^2. We can describe families of “elementary" links which have such surgeries, and one can ask whether all links with few components having such surgeries lie in these families. The answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer for a special family of 2-component links. This is joint work with Rob Kirby. Gay and Kirby formulated a new way to decompose a (closed, orientable) 4-manifold M, called a trisection. I’ll describe how to translate from a classical framed link diagram for M to a trisection diagram. The links so obtained lie on Heegaard surfaces in the 3-sphere, and have surgeries yielding some number of copies of S^1XS^2. We can describe families of “elementary" links which have such surgeries, and one can ask whether all links with few components having such surgeries lie in these families. The answer is almost certainly no. We nevertheless give a small piece of evidence in favor of a positive answer for a special family of 2-component links. This is joint work with Rob Kirby.

Mon, 14 Feb 2022

15:30

15:30

L5

Andrea Seppi

(University of Grenoble-Alpes)

Minimal Lagrangian maps play an important role in Teichmüller theory, with important existence and uniqueness results for hyperbolic surfaces obtained by Labourie, Schoen, Bonsante-Schlenker, Toulisse and others. In positive curvature, it is thus natural to ask whether one can find minimal Lagrangian diffeomorphisms between two spherical surfaces with cone points. In this talk we will show that the answer is negative, unless the two surfaces are isometric. As an application, we obtain a generalization of Liebmann’s theorem for branched immersions of constant curvature in Euclidean space. This is joint work with Christian El Emam.

Mon, 07 Feb 2022

15:30

15:30

C3

Naomi Andrew

(Southampton University)

Free-by-cyclic groups are easy to define – all you need is an automorphism of F_n. Their properties (for example hyperbolicity, or relative hyperbolicity) depend on this defining automorphism, but not always transparently. I will introduce these groups and some of their properties, and connect some to properties of the defining automorphism. I'll then discuss some ideas and techniques we can use to understand their automorphisms, including finding useful actions on trees and relationships with certain subgroups of Out(F_n). (This is joint work with Armando Martino.)

Mon, 31 Jan 2022

15:30

15:30

Virtual

Rufus Willett

(Hawaii)

Let X be a closed Riemannian manifold, and represent the algebra C(X) of continuous functions on X on the Hilbert space L^2(X) by multiplication. Inspired by the heat kernel proof of the Atiyah-Singer index theorem, I'll explain how to describe K-homology (i.e. the dual theory to Atiyah-Hirzebruch K-theory) in terms of parametrized families of operators on L^2(X) that get more and more 'local' in X as time tends to infinity.

I'll then switch perspectives from C(X) -- the prototypical example of a commutative C*-algebra -- to noncommutative C*-algebras coming from discrete groups, and explain how the underlying large-scale geometry of the groups can give rise to approximate 'decompositions' of the C*-algebras. I'll then explain how to use these decompositions and localization in the sense above to compute K-homology, and the connection to some conjectures in topology, geometry, and C*-algebra theory.

Mon, 24 Jan 2022

15:30

15:30

Virtual

Lukas Brantner

(Oxford)

Over the complex numbers, the Bomolgorov-Tian-Todorev theorem asserts that Calabi-Yau varieties have unobstructed deformations, so any n^{th} order deformation extends to higher order. We prove an analogue of this statement for the nicest kind of Calabi-Yau varieties in characteristic p, namely ordinary ones, using derived algebraic geometry. In fact, we produce canonical lifts to characteristic zero, thereby generalising results of Serre-Tate, Deligne-Nygaard, Ward, and Achinger-Zdanowic. This is joint work with Taelman.

Mon, 17 Jan 2022

15:30 -
16:30

Virtual

Ian Zemke

(Princeton)

Lattice homology is a combinatorial invariant of plumbed 3-manifolds due to Nemethi. The definition is a formalization of Ozsvath and Szabo's computation of the Heegaard Floer homology of plumbed 3-manifolds. Nemethi conjectured that lattice homology is isomorphic to Heegaard Floer homology. For a restricted class of plumbings, this isomorphism is known to hold, due to work of Ozsvath-Szabo, Nemethi, and Ozsvath-Stipsicz-Szabo. By using the Manolescu-Ozsvath link surgery formula for Heegaard Floer homology, we prove the conjectured isomorphism in general. In this talk, we will talk about aspects of the proof, and some related topics and extensions of the result.

Mon, 29 Nov 2021

15:45

15:45

Virtual

Zoltan Szabo

(Princeton University)

In a joint work with Peter Ozsvath we have developed algebraic invariants for knots using a family of bordered knot algebras. The goal of this lecture is to review these constructions and discuss some of the latest developments.

Mon, 22 Nov 2021

15:45

15:45

Virtual

Emily Stark

(Wesleyan University)

Rigidity theorems prove that a group's geometry determines its algebra, typically up to virtual isomorphism. Motivated by rigidity problems, we study graphically discrete groups, which impose a discreteness criterion on the automorphism group of any graph the group acts on geometrically. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds. We will present new examples, proving this property is not a quasi-isometry invariant. We will discuss action rigidity for free products of residually finite graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

Mon, 15 Nov 2021

15:45

15:45

Virtual

Bruno Martelli

(Universita di Pisa)

We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions $n\le 5$. As a consequence, there are hyperbolic groups with finite-type subgroups that are not hyperbolic.

The main tool is Bestvina - Brady theory enriched with a combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

Mon, 08 Nov 2021

15:45

15:45

Virtual

Shintaro Nishikawa

(University of Münster)

Consider simple rank-one Lie groups $SO(n, 1)$, $SU(n, 1)$ and $Sp(n ,1)$ ($n>1$). They are the isometry groups of real, complex and quaternionic hyperbolic spaces respectively.

By a result of Kostant, the trivial representation of $Sp(n ,1)$ is isolated in the space of irreducible unitary representations on Hilbert spaces. That is, $Sp(n ,1)$ has Kazhdan’s property (T) which is equivalent to the vanishing of 1st cohomology of the group in all unitary representations. This is in contrast to the case of $SO(n ,1)$ and $SU(n ,1)$ where they have the Haagerup approximation property, a strong negation of property (T).

This dichotomy between $SO(n ,1)$, $SU(n ,1)$ and $Sp(n ,1)$ disappears when we consider so-called uniformly bounded representations on Hilbert spaces. By a result of Cowling in 1980’s, the trivial representation of $Sp(n ,1)$ is no longer isolated in the space of uniformly bounded representations. Moreover, there is a uniformly bounded representation of $Sp(n ,1)$ with non-zero first cohomology group.

The goal of this talk is to describe these facts.

Mon, 01 Nov 2021

15:45

15:45

Virtual

Joshua Greene

(Boston College)

I will talk about joint work with Andrew Lobb related to Toeplitz's square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.

Mon, 25 Oct 2021

15:45

15:45

Virtual

Daniel Berwick Evans

(University of Illinois at Urbana-Champaign)

Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric) field theories. Properties of these field theories lead to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from the field theory point of view. In this talk, we will describe a map from 2-dimensional field theories to a cohomology theory that approximates TMF. This map affords a cocycle description of certain torsion classes. In particular, we will explain how a choice of anomaly cancelation for the supersymmetric sigma model with target $S^3$ determines a cocycle representative of the generator of $\pi_3(TMF)=\mathbb{Z}/24$.

Mon, 18 Oct 2021

15:45

15:45

Virtual

Arman Darbinyan

(Texas A&M)

Topologically speaking, left-orderable countable groups are precisely those countable groups that embed into the group of orientation preserving homeomorphisms of the real line. A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. We will also discuss some of the algebraic, geometric, and computability properties that this construction bears. The construction is based on novel topological and geometric methods that also will be discussed. The flexibility of the embedding method allows us to go beyond the class of left-orderable groups as well. Based on a joint work with Markus Steenbock.

Mon, 11 Oct 2021

15:45

15:45

L4

Sam Hughes

(Oxford University)

In this talk we will introduce Leary and Minasyan's CAT(0) but not biautomatic groups as lattices in a product of a Euclidean space and a tree. We will then investigate properties of general lattices in that product space. We will also consider a construction of lattices in a Salvetti complex for a right-angled Artin group and a Euclidean space. Finally, if time permits we will also discuss a "hyperbolic Leary–Minasyan group" and some work in progress with Motiejus Valiunas towards an application.

Mon, 14 Jun 2021

15:45 -
16:45

Virtual

Ana Lecuona

(University of Glasgow)

Together with Alex Degtyarev and Vincent Florence we introduced a new link invariant, called slope, of a colored link in an integral homology sphere. In this talk I will define the invariant, highlight some of its most interesting properties as well as its relationship to Conway polynomials and to the Kojima–Yamasaki eta-function. The stress in this talk will be on our latest computational progress: a formula to calculate the slope from a C-complex.

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