# Topology Festival

May 10–12, 2019

## Abstracts of Talks (2018)

### Normal Subgroups of Mapping Class Groups

The mapping class group of a surface has an incredibly rich normal subgroup structure. For this reason, a traditional classification theorem for normal subgroups of mapping class groups in the form of a complete list of isomorphism types is almost certainly out of reach. In this talk, we will discuss recent work with Margalit giving an easy-to-check criterion for a normal subgroup to have the extended mapping class group as its automorphism group. This result applies to a wide class of normal subgroups and enables us to use the automorphism group as an invariant of normal subgroups, suggesting a new framework for the classification of normal subgroups of mapping class groups.

### New Methods for Finite Generation

Automorphisms of surface groups and free groups are fundamental in low-dimensional topology and have a close analogy with arithmetic groups like $GL_n(Z)$.  A foundational question is whether the analogue of “congruence subgroups” are finitely generated. For the first level, this was proved by Dehn (1938); for the second level, it was proved by Johnson (1983). McCullough-Miller conjectured in 1986 that the third level should NOT be finitely generated, but this remained open until this year. (A proof was published in 2006, but turned out to be flawed.)

We introduce a new method for proving a group is finitely generated, and use this to disprove this conjecture for ALL levels: At every level, the congruence subgroups are finitely generated. Joint work with Mikhail Ershov and Andrew Putman, building on work of Mikhail Ershov and Sue He.

### Hochschild-Mitchell Homology of Stratified Linear Categories

Hochschild-Mitchell homology $HH_*(C)$ of  a linear category $C$ is one of some fundamental invariants of linear categories.  A stratified linear category is a linear category $C$ equipped with a distinguished linear subcategory $C^0$ and a filtration on the set of objects such that the Hom spaces satisfy a certain factorization property.  Then $HH_*(C)$ is isomorphic to $HH_*(C^0)$.  A similar result holds for cyclic homology as well. One can apply this result to the categorified quantum $sl(2)$, for which we computed the $HH_0$ in a joint work with Beliakova, Lauda, and Zivkovic. I plan to explain some other examples as well.

### Braided Tensor Categories and the Cobordism Hypothesis

According to the cobordism hypothesis, a fully local $n$-dimensional TFT with values in some $(\infty,n)$ category is determined by its “fully dualizable” and “$SO(n)$-fixed” objects.  I will recall this general framework, and then explain recent works of R. Haugseng, T. Johnson-Freyd and C. Scheimbauer, which builds a 4-category called the Morita theory of braided tensor categories.  Finally, I'll describe recent work of mine with A. Brochier and N. Snyder identifying natural 3- and 4-dualizable subcategories thereof: the rigid (3-dualizable) and fusion (4-dualizable) braided tensor categories.  I'll also report on work in progress, also joint with Brochier and Snyder, constructing $SO(3)$- and $SO(4)$-fixed points, from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain new fully local 3- and 4-dimensional topological field theories.  I'll outline expected relationships of these to: Crane-Yetter-Kauffman invariants, quantum A-polynomials, and DAHA-Jones polynomials.

### Equivariant Cohomology Distinguishes Circle Actions on a Symplectic Four-Manifold

Hamiltonian $S^1$-spaces of dimension four are classified by the associated decorated graphs. We give a generators and relations description of the equivariant cohomology of a Hamiltonian $S^1$-space. We use the description to show that the equivariant cohomology determines the decorated graph, sans the heights and area labels, up to flipping chains. As a result, we get a proof of the finiteness of Hamiltonian circle actions on a closed symplectic four-manifold, that does not use pseudo-holomorphic tools.

This is a joint work with Tara Holm.

### Categorification in Topology and Representation Theory

Categorification of topological invariants allows to go one dimension up in the hierarchical ladder. On the algebraic side, it corresponds to going from semisimple to triangulated categories and leads to new connections and an enriched viewpoint on representation theory and homological algebra. These structures will be reviewed in the talk.

### How to Category the Ring of Integers with Two Inverted

We will explain how to combine diagrammatic and algebraic constructions to produce a monoidal triangulated category with the Grothendieck ring naturally isomorphic to the ring of integers with inverted two. The talk is based on the joint work with Yin Tian.

### Representation Homology of Spaces

Representation homology of topological spaces is a natural homological extension of the representation varieties of fundamental groups. In this talk, we will give an interpretation of representation homology as functor homology and explain its relation to other known homology theories associated with spaces (such as higher Hochschild homology, Pontryagin algebras and $S^1$-equivariant homology of free loop spaces). One of our main technical results is a computation of the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Quillen and Sullivan models. We will also compute explicitly the representation homology of some interesting non-simply connected spaces, such as Riemann surfaces and link complements in $R^3$. In the case of link complements, we obtain a new homological invariant of links analogous to knot contact homology. Time permitting, we will also discuss some applications to representation theory, including an intriguing relation to the celebrated Strong Macdonald Conjecture of Macdonald, Feigin and Hanlon.

This is joint work with Yuri Berest and Wai-kit Yeung.

### Coalgebras, coTHH, and K-Theory

I will discuss joint work with Hess and other coauthors studying (topological) coHochschild homology for differential graded coalgebras and for coalgebra spectra and connections to K-theory.

### Introduction to TQFTs and HQFTs (Friday)

I discuss geometric and algebraic ideas underlying the available constructions of topological quantum field theories and homotopy quantum field theories.

### Brackets, Cobrackets, and Double Brackets in the World of Loops (Saturday)

In this talk I will survey geometric constructions on loops which lead to remarkable algebraic structures in homology of the loop spaces of manifolds:  Lie brackets, Lie cobrackets, double brackets, etc.

### Representations of Kauffman Bracket Skein Algebras of a Surface

The definition of the Kauffman bracket skein algebra of an oriented surface was originally motivated by the Jones polynomial, and a representation of the skein algebra features importantly in the topological quantum field theory description of Witten's 3-manifold invariant.  Later, the skein algebra of a surface was found to bear deep relationships to hyperbolic geometry, via the SL2C-character variety.  We will explicate this relationship between the skein algebra and the hyperbolic geometry of a surface, using representations.