Past Topology Seminar

In this talk we will discuss the connection between combinatorial properties of minimally self-intersecting curves on a surface S and the geometric behaviour of geodesics on S when S is endowed with a Riemannian metric. In particular, we will explain the interplay between a smoothing, which is a type of surgery on a curve that resolves a self-intersection, and k-systoles, which are shortest geodesics having at least k self-intersections, and we will present some results that partially elucidate this interplay. There will be lots of pictures. Based on joint work with Max Neumann-Coto.

Introduced by Bestvina and Brady in 1997, Bestvina—Brady groups form an important class of examples in geometric group theory and topology, known for exhibiting unusual finiteness properties. In this talk, I will focus on the Dehn functions of finitely presented Bestvina—Brady groups. Very roughly speaking, the Dehn function of a group measures how difficult it is to fill loops by discs in spaces associated to the group, and captures geometric information that is invariant under coarse equivalence. After reviewing known results, I will present a classification of the Dehn functions of Bestvina—Brady groups. This talk is based on joint work with Yu-Chan Chang and Matteo Migliorini.

I will discuss a uniform waist inequality in codimension 2 for the family of finite covers of a Riemannian manifold whose fundamental group has Kazhdan‘s property T. I will describe a general strategy to prove waist inequalities based on a higher property T for Banach spaces. The general strategy can be implemented in codimension 2 but is conjectural in higher codimension. We speculate about the situation for lattices in semisimple Lie groups. Based on joint work with Uri Bader

The spectral gap (or bass note) of a closed hyperbolic surface is the smallest non-zero eigenvalue of its Laplacian. This invariant plays an important role in many parts of hyperbolic geometry. In this talk, I will speak about joint work with Will Hide on the question of which numbers can appear as spectral gaps of closed arithmetic hyperbolic surfaces.

I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the congruence subgroup property, relates the finite-index subgroups of mapping class groups to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. Time permitting, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.

In the 1990s, Goodwillie developed a theory of calculus for homotopical functors. His idea was to approximate a functor by a tower of ‘polynomial functors’, similar to how one approximates a function by its Taylor series. The role of linear polynomials is played by functors that behave like homology theories, in the sense that there is a Mayer-Vietoris sequence computing their homotopy groups. As such, the Goodwillie tower interpolates between stable and unstable homotopy theory. The theory has application to the computation of the homotopy groups of spheres, higher algebra, and algebraic K-theory. In my talk, I will give an introduction to this topic. In particular, I will explain that Goodwillie's calculus reveals a deep connection between the homotopy theory of spaces and Lie algebras and how this is related to a chain rule for the derivatives of functors.
 

I will talk about the problem of recognising when a manifold admits a finite cover that fibres over the circle, with emphasis on the case of hyperbolic manifolds in odd dimensions. I will survey the state-of-art, and discuss the role that group theory plays in the problem. Finally, I will discuss a recent result that sheds light on the analogous group-theoretic problem, that is, virtual algebraic fibring of Poincaré-duality groups. The final theorem is joint with Sam Fisher and Giovanni Italiano.

Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, (the better) half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism that can be viewed as mixing sheaf theory with functional analysis. Based on joint work with André Henriques.

Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras (homology is here taken to be certain Tor-groups), when Boyd and Hepworth showed that in low dimensions the homology vanishes. We're now able to give complete calculations of their homology, which has a surprisingly rich structure (and in particular is very far from vanishing). This is joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka. Prerequisites will be minimal: it will be enough to know what Tor is.

Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic 3-manifolds and the main result will replace a spectral gap problem related to torsion in homology with a geometric version involving geodesic length and stable commutator length. As an application, we provide "bad" examples of hyperbolic 3-manifolds with bounded geometry but extremely small (1-form) spectral gaps.

I will discuss an enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. Using these characterizations, it follows that an enriched finite braided tensor category is invertible as an object of the Morita 4-category of enriched braided tensor categories if and only if it is non-degenerate. As an application, I will explain how to extend the full dualizability result of Brochier-Jordan-Synder by showing that a finite braided tensor category is fully dualizable in the Morita 4-category of braided tensor categories if its symmetric center is separable.
 

Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of $S^n \times  S^n$. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of $S^2 \times  S^2$. This talk is based on joint work with John Nicholson and Simona Veselá.

Given a triangulated 3-manifold, can we decide whether it is hyperbolic? In general, no efficient algorithm for answering this question is known; however, the problem becomes more manageable if we restrict our attention to specific classes of 3-manifolds. In this talk, I will discuss how to certify that a triangulated fibred 3-manifold is hyperbolic, in polynomial time in the size of the triangulation and in the Euler characteristic of the fibre. The argument relies on the theory of normal surfaces, as well as several previously known certification algorithms, of which I will give a survey. I will also mention, time permitting, a recent algorithm to decide if an element of the mapping class group of a surface is pseudo-Anosov in polynomial time, which is used in the certification procedure.

We study the spectrum of the Laplacian on finite-area hyperbolic surfaces of large volume, focusing on small eigenvalues i.e. those below 1/4. I will discuss some recent results and open problems in this area. Based on joint works with Michael Magee and with Joe Thomas.
 

Gauge theory excels at solving minimal genus problems for 3- and 4-manifolds.  A notable triumph is its resolution of the Thom conjecture, asserting that the genus of a smooth complex curve in the complex projective plane is no larger than any smooth submanifold homologous to it.  Gauge theoretic techniques have also been used to verify analagous conjectures for Kähler surfaces or, more generally, symplectic 4-manifolds.  One can formulate versions of these conjectures for surfaces with boundary lying in a 3-manifold, and I'll discuss work in progress with Katherine Raoux which attempts to extend these "relative" Thom conjectures outside the complex (or even symplectic) realm using tools from Floer homology.

The Farrell--Jones conjecture predicts that the algebraic K-theory of a group ring is isomorphic to a certain equivariant homology theory, and there are also versions for L-theory and Waldhausen's A-theory. In principle, this provides a way to calculate these K-groups, and has many applications. These include classifying manifolds admitting a given fundamental group and a positive resolution of the Borel conjecture.

I will discuss work with Yassine Guerch and Sam Hughes on the Farrell--Jones conjecture for extensions of relatively hyperbolic groups, as well as an application to their automorphism groups in the one-ended case. The methods are from geometric group theory: we go via the theory of JSJ decompositions to produce acylindrical actions on trees.

I want to give a survey about the rational cohomology of SL_n
Z. This includes recent developments of finding Hopf algebras in the
direct sum of all cohomology groups of SL_n Z for all n. I will give a
quick overview about Hopf algebras and what this structure implies for
the cohomology of SL_n Z.

Topological Quantum Field Theories (TQFTs) have been studied as mathematical toy models for quantum field theories in physics and are described by a functor out of some bordism category. In dimension 2, TQFTs are fully classified by Frobenius algebras. Homotopy Quantum Field Theories (HQFTs), introduced by Turaev, consider additional homotopy data to some target space X on the bordism categories. For homotopy 1-types Turaev also gives a classification via crossed G-Frobenius algebras, where G denotes the fundamental group of X.
In this talk we will introduce a multi-object generalization of Frobenius algebras called Frobenius categories and give a version of this classification theorem involving the fundamental groupoid. Further, we will give a classification theorem for HQFTs with target homotopy 2-types by considering crossed modules (joint work with Alexis Virelizier).
 

June Huh proved in 2012 that the Betti numbers of the complement of a complex hyperplane arrangement form a log concave sequence.  But what if the arrangement has symmetries, and we regard the cohomology as a representation of the symmetry group?  The motivating example is the braid arrangement, where the complement is the configuration space of n points in the plane, and the symmetric group acts by permuting the points.  I will present an equivariant log concavity conjecture, and show that one can use representation stability to prove infinitely many cases of this conjecture for configuration spaces.
 

In general, the classification of finitely generated subgroups of a given group is intractable. Restricting to two-generator subgroups in a geometric setting is an exception. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. While the statement is algebraic, the proof technique uses the topology of finite graphs, a la Stallings. This is joint work with Naomi Andrew, Ilya Kapovich, and Stefano Vidussi.
 

We show that linear reflection groups in the sense of Vinberg are often Zariski dense in PGL(n). Among the applications are examples of low-dimensional closed hyperbolic manifolds whose fundamental groups virtually embed as Zariski-dense subgroups of SL(n,Z), as well as some one-ended Zariski-dense subgroups of SL(n,Z) that are finitely generated but infinitely presented, for all sufficiently large n. This is joint work with Jacques Audibert, Gye-Seon Lee, and Ludovic Marquis.

I will talk about certain higher algebraic structure, governed by Kontsevich's Lie graph complex, that can be associated to an oriented fibration with Poincaré duality fiber. We construct a generalized fiber integration map associated to each Lie graph homology class and the main result is that this gives a faithful representation of graph homology. I will discuss how this leads to new possible interpretations of Lie graph homology classes as obstructions to, on one hand, smoothness of Poincaré duality fibrations, and, on the other hand, the existence of Poincaré duality algebra resolutions of the cochains of the total space as a dg module over the cochains of the base space.

In his final paper in 1954, Alan Turing wrote `No systematic method is yet known by which one can tell whether two knots are the same.' Within the next 20 years, Wolfgang Haken and Geoffrey Hemion had discovered such a method. However, the computational complexity of this problem remains unknown. In my talk, I will give a survey on this area, that draws on the work of many low-dimensional topologists and geometers. Unfortunately, the current upper bounds on the computational complexity of the knot equivalence problem remain quite poor. However, there are some recent results indicating that, perhaps, knots are more tractable than they first seem. Specifically, I will explain a theorem that provides, for each knot type K, a polynomial p_K with the property that any two diagrams of K with n_1 and n_2 crossings differ by at most p_K(n_1) + p_K(n_2) Reidemeister moves.

 A commensuration of a group G is an isomorphism between finite-index subgroups of G. Equivalence classes of such maps form a group, whose importance first emerged in the work of Margulis on the rigidity and arithmeticity of lattices in semisimple Lie groups. Drawing motivation from this classical setting and from the study of mapping class groups of surfaces, I shall explain why, when N is at least 3, the group of automorphisms of the free group of rank N is its own abstract commensurator. Similar results hold for certain subgroups of Aut(F_N). These results are the outcome of a long-running project with Ric Wade. An important element in the proof is a non-abelian analogue of the Fundamental Theorem of Projective Geometry in which projective subspaces are replaced by the free factors of a free group; this is the content of a long-running project with Mladen Bestvina.
 

The simplicial volume of a simplicial complex is a topological invariant
related to the growth of the fundamental group, which gives rise to a
semi-norm in homology. In this talk, we introduce the volume entropy
semi-norm, which is also related to the growth of the fundamental group
of simplicial complexes and shares functorial properties with the
simplicial volume. Answering a question of Gromov, we prove that the
volume entropy semi-norm is equivalent to the simplicial volume
semi-norm in every dimension. Joint work with I. Babenko.
 

I will talk about link and three-manifold invariants defined in terms of a non-semisimple finite ribbon category C together with a choice of tensor ideal and a trace on it. If the ideal is all of C, these invariants agree with those defined by Lyubashenko in the 90’s, and as we show, they only depend on the Grothendieck class of the objects labelling the link. These invariants are therefore not able to determine non-split extensions, or they are algebraically weak. However, we observed an interesting phenomenon: if one chooses an intermediate proper ideal between C and the minimal ideal of projective objects, the invariants become algebraically much stronger because they do distinguish non-trivial extensions. This is demonstrated in the case of C being the super-modular category of an exterior algebra. That is why these invariants deserve to be called “non-semisimple”. This is a joint work with J. Berger and I. Runkel.

We will discuss the problem of finding hyperbolic manifolds fibring over the circle; and show a method to construct and analyse maps from particular hyperbolic manifolds to S^1, which relies on Bestvina-Brady Morse theory. 
This technique can be used to build and detect fibrations, algebraic fibrations, and Morse functions with minimal number of critical points, which are interesting in the even dimensional case. 
After an introduction to the problem, and presentation of the main results, we will use the remaining time to focus on some easy 3-dimensional examples, in order to explicitly show the construction at work.
 

The objects of arithmetic geometry are not manifolds. Some concepts from differential geometry admit analogues in arithmetic, but they are not straightforward. Nevertheless, there is a growing sense that the right way to understand certain Langlands phenomena is to study quantum field theories on these objects. What hope is there of making this thought precise? I will propose the beginnings of a mathematical framework via a general theory of factorization algebras. A new feature is a subtle piece of additional structure on our objects – what I call an _isolability structure_ – that is ordinarily left implicit.

It is a classical fact of rational homotopy theory that the $E_\infty$-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still $E_n$-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of $E_n$-algebras, closely related to Koszul duality for the $E_n$-operads. We will also see that these results are essentially sharp: if the R-valued cochains of $S^n$ have square-zero $E_{n+1}$-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

I will discuss three different constructions of smooth tori in S^4 whose complements have fundamental group Z: turned 1-twist-spun tori due to Boyle, the union of a ribbon disc with a genus one Seifert surface constructed by Cochran and Davis, and certain tori with four critical points. They are all topologically unknotted, but it is not known whether they are smoothly standard, except for tori with four critical points whose middle level set is a split link. The branched double cover of S^4 along any of these surfaces is a potentially exotic copy of S^2 x S^2, though, in the case of Boyle's example, it cannot be distinguished from the standard S^2 x S^2 using Seiberg-Witten invariants. This is joint work with Mark Powell.

A subset in the ideal boundary of a CAT(0) space is called symmetric if every complete geodesic with one ideal boundary point
in the set has both ideal boundary points in the set. In the late 80s Eberlein proved that if a Hadamard manifold contains a non-trivial closed symmetric  subset in its ideal boundary, then its holonomy group cannot act transitively. This leads to rigidty via
the Berger-Simons Theorem. I will discuss rigidity of ideal symmetric sets in the general context of locally compact geodesically complete
CAT(0) spaces.
 

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.

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.

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.

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.