Past Topology Seminar

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.
 

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.

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.
 

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. 

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).

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.

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.

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.  

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.   

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.

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.

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.

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.
 

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.
 

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.

I shall begin with a brief history of the problem of trying to understand infinite groups knowing only their finite quotients. I'll then focus on 3-manifold groups, describing the prominent role that they have played in advancing our understanding of this problem in recent years. The story for 3-manifold groups involves a rich interplay of algebra, geometry, and arithmetic. I shall describe arithmetic Kleinian groups that are profinitely rigid in the absolute sense -- ie they can be distinguished from all other finitely generated, residually finite groups by their set of finite quotients. I shall then explain more recent work involving products of Seifert fibered manifolds -- here we find groups that are profinitely rigid in the class of finitely presented groups but not in the class of finitely generated groups. This is joint work with McReynolds, Reid, and Spitler.

I shall discuss my recent work showing that the Bogomolov-Tschinkel universality conjecture holds if and only if the mapping class groups of a punctured surface is large. One consequence of this result is that all genus 2 surface-by-surface (and all genus 2 surface-by-free) groups are virtually algebraically fibered. Moreover, I will explain why simple curve homology does not always generate homology of finite covers of closed surface. I will also mention my work with O. Tosic regarding the Putman-Wieland conjecture, and explain the partial solution to the Prill's problem about algebraic curves.

 

The chromatic polynomial \chi(Q) can be defined for any graph, such that for Q integer it counts the number of colourings. It has many remarkable properties, and I describe several that are derived easily by using fusion categories, familiar from topological quantum field theory. In particular, I define the chromatic algebra, a planar algebra whose evaluation gives the chromatic polynomial. Linear identities of the chromatic polynomial at certain values of Q then follow from the Jones-Wenzl projector of the associated category. An unusual non-linear one called Tutte's golden identity relates \chi(\phi+2) for planar triangulations to the square of \chi(\phi+1), where \phi is the golden mean. Tutte's original proof is purely combinatorial. I will give here an elementary proof by manipulations of a topological invariant related to the Jones polynomial. Time permitting, I will also mention analogous identities for graphs on more general surfaces. Based on work with Slava Krushkal.

A modular functor is defined as a system of mapping class group representations on vector spaces (the so-called conformal blocks) that is compatible with the gluing of surfaces. The notion plays an important role in the representation theory of quantum groups and conformal field theory. In my talk, I will give an introduction to the theory of modular functors and recall some classical constructions. Afterwards, I will explain the approach to modular functors via cyclic and modular operads and their bicategorical algebras. This will allow us to extend the known constructions of modular functors and to classify modular functors by certain cyclic algebras over the little disk operad for which an obstruction formulated in terms of factorization homology vanishes. (The talk is based to a different extent on different joint works with Adrien Brochier, Lukas Müller and Christoph Schweigert.)

Around 30 years ago, Lin defined an analog of the Casson invariant for knots. This invariant counts representations of the knot group into SU(2) which satisfy tr(ρ(m)) = c for some fixed c. As a function of c, the Casson-Lin invariant turns out to be given by the Levine-Tristram signature function.

If K is a small knot in S³, I'll describe a version of the Casson-Lin invariant which counts representations of the knot group into SL₂(R) with tr(ρ(m)) = c for c in [-2,2]. The sum of the SU(2) and SL₂(R) invariants is a constant h(K), independent of c. I'll discuss the proof of this fact and give some applications to the existence of real parabolic representations and left-orderings. This is joint work with Nathan Dunfield.

THIS TALK IS CANCELLED DUE TO ILLNESS -- In this talk I will present classification results for rigid Frobenius algebras in Dijkgraaf–Witten categories ℨ( Vec(G)ᵚ ) over a field of arbitrary characteristic, generalising existing results by Davydov-Simmons. For this purpose, we provide a braided Frobenius monoidal functor from ℨ ( Vect(H)ᵚˡᴴ ) to ℨ( Vec(G)ᵚ ) for any subgroup H of G. I will also discuss about their categories of local modules, which are modular tensor categories  by results of Kirillov–Ostrik in the semisimple case and Laugwitz–Walton in the general case. Joint work with Robert Laugwitz (Nottingham) and Sam Hannah (Cardiff).

Ideal triangulations were introduced by Thurston as a tool for studying hyperbolic three-manifolds.  Taut ideal triangulations were introduced by Lackenby as a tool for studying "optimal" representatives of second homology classes.

After these applications in geometry and topology, it is time for dynamics. Veering triangulations (taut ideal triangulations with certain decorations) were introduced by Agol to study the mapping tori of pseudo-Anosov homeomorphisms.  Gueritaud gave an alternative construction, and then Agol and Gueritaud generalised it to find veering triangulations of three-manifolds admitting pseudo-Anosov flows (without perfect fits).

We prove the converse of their result: that is, from any veering triangulation we produce a canonical dynamic pair of branched surfaces (in the sense of Mosher).  These give flows on appropriate Dehn fillings of the original manifold.  Furthermore, our construction and that of Agol--Gueritaud are inverses.  This then gives a "perfect" combinatorialisation of pseudo-Anosov flow (without perfect fits).

This is joint work with Henry Segerman.

In this talk, I will first give a brief introduction to the Landau-Ginzburg -- Conformal Field Theory (LG-CFT) correspondence, a prediction from physics. This prediction links aspects of Landau-Ginzburg models, described by matrix factorisations for a polynomial known as the potential, with Conformal Field Theories, described by for example vertex operator algebras. While both sides of the correspondence have good mathematical descriptions, it is an open problem to give a mathematical formulation of the correspondence. 

After this introduction, I will discuss the only known realisation of this correspondence, for the potential $x^d$. For even $d$ this is a recent result, and I will give a sketch of the proof which uses the tools of module tensor categories

 I will not assume prior knowledge of matrix factorisations, CFTs, or module tensor categories. This talk is based on joint work with Ana Ros Camacho.

Two CW-complexes are simple homotopy equivalent if they are related by a sequence of collapses and expansions of cells. It implies homotopy equivalent as is implied by homeomorphic. This notion proved extremely useful in manifold topology and is central to the classification of non-simply connected manifolds up to homeomorphism. I will present the first examples of two 4-manifolds which are homotopy equivalent but not simple homotopy equivalent, as well as in all higher even dimensions. The examples are constructed using surgery theory and the s-cobordism theorem, and are distinguished using methods from algebraic number theory and algebraic K-theory. I will also discuss a number of new directions including progress on classifying the possible fundamental groups for which examples exist. This is joint work with Csaba Nagy and Mark Powell.

I will present classification results for 4-manifolds with boundary and infinite cyclic fundamental group, obtained in joint work with Anthony Conway and with Conway and Lisa Piccirillo.  Time permitting, I will describe applications to knotted surfaces in simply connected 4-manifolds, and to investigating the difference between the relations of homotopy equivalence and stable homeomorphism. These will also draw on work with Patrick Orson and with Conway,  Diarmuid Crowley, and Joerg Sixt.

Bestvina--Feighn proved that Aut(F_n) is a rational duality group, i.e. there is a Q[Aut(F_n)]-module, called the rational dualizing module, and a form of Poincare duality relating the rational cohomology of Aut(F_n) to its homology with coefficients in this module. Bestvina--Feighn's proof does not give an explicit combinatorial description of the rational dualizing module of Aut(F_n). But, inspired by Borel--Serre's description of the rational dualizing module of arithmetic groups, Hatcher--Vogtmann constructed an analogous module for Aut(F_n) and asked if it is the rational dualizing module. In work with Miller, Nariman, and Putman, we show that Hatcher--Vogtmann's module is not the rational dualizing module.

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.

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.

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.

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.

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.


 

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.

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.

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

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.

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).

 

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.  

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.

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.)

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.

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.

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.

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.

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.

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.

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.

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.

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 $\pi_3(TMF)=\mathbb{Z}/24$.

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.

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.