Purdue Topology Seminar, Fall 2019

Abstract: Since the discovery of the Jones polynomial and the formulation of the topological quantum field theories (TQFTs) in the 1980s, there has been significant progress in the area now called quantum algebra/topology. One important application of TQFTs in topology is that they provide quantum invariants of smooth manifolds. Quantum invariants of 3-manifolds have been studied extensively, among which the Reshetikhin-Turaev invariants and the Turaev-Viro invariants form two fundamental families. On the other hand, quantum invariants of 4-manifolds are much less well-understood.

We give a construction of a family of invariants of 4-manifolds that can be viewed as a 4-dimensional analog of the 3D Kuperberg invariant which in turn is a generalization of the Turaev-Viro invariant. The algebraic data for the construction is what we call a Hopf triple which consists of three Hopf algebras and a bilinear form on each pair of the Hopf algebras satisfying certain compatibility conditions. Quasi-triangular Hopf algebras are special examples of Hopf triples. Trisection diagrams of 4-manifolds recently proposed by Gay and Kirby will be used in the construction of the invariant. Some interesting properties of the invariant will also be discussed.

In the first part (2:30-3:15pm), I’ll give a basic introduction to TQFTs and quantum invariants. The main talk (3:30-4:30pm) will mostly focus on the 4-dimensional quantum invariants.

Abstract: One of the main goals in algebraic topology is to construct algebraic invariants for a class of spaces that determine those spaces up to a notion of equivalence, such as homeomorphism, homotopy equivalence, or some weaker notion. M. Mandell has proven that certain algebraic structure associated to spaces, known as the “E-infinity coalgebra of singular chains on the space”, completely classifies simply connected spaces up to weak homotopy equivalence. In this talk I will discuss a key point that allows to extend this result to all path connected spaces. This key point is an interesting and fundamental fact by itself, it can be discussed without previous knowledge of Mandell’s results, and opens up new directions and questions in algebraic topology.

More precisely, the key point says that the fundamental group of a path connected space, as well as higher dimensional homotopical aspects, can be determined, in complete generality, directly from algebraic structure of the singular chains on the underlying space. There are three main ingredients that come into play in order to give a transparent and precise formulation of this fact: 1) we extend a classical result of F. Adams from 1956 regarding an algebraic model (called the “cobar construction”) for the based loop space of a simply connected space, 2) we make use of the homotopical symmetry of the chain approximations to the diagonal map on a space, and 3) we apply a duality theory for algebraic structures known as Koszul duality.

The first 45 minutes (2:30pm-3:15pm) will be introductory and accesible to a general audience with a basic knowledge of topology. This introduction will also serve as preparation for the main talk (3:30pm-4:30pm) in which I will describe the main constructions and proofs in more detail. This is all joint work with Mahmoud Zeinalian and Felix Wierstra.

Abstract: The (co)homology groups of GLn(Z) are important in algebraic topology and number theory. By the work of
Borel, Voevodksy, Weibel, and others, these groups are well understood in a stable range. In this talk, I will discuss joint
work in progress with Alexander Kupers, Peter Patzt, Rohit Nagpal, and Jennifer Wilson on improvements to this stable
range.

Title (PP): Top dimensional cohomology of principal congruence subgroups

Abstract: The level p congruence subgroup of SL_n(Z) is defined to be the subgroup of matrices congruent to the identity matrix
mod p. These groups have trivial cohomology in high enough degrees. In the 1970s, Lee and Szczarba gave a conjectural
description of the top cohomology groups of these congruence subgroups. In joint work in progress with Jeremy Miller and
Andrew Putman, we show that this conjecture is false and that these congruence subgroups have extra exotic cohomology
classes in their top degree cohomology coming from the first homology group of the associated modular curve.

Intro talk abstract: I’ll give an overview of some basic ideas from computational complexity theory, and what the big picture is for the complexity of various problems in and around 3-manifold topology.

Main talk title: Coloring invariants of knots are often intractable

Abstract: Coloring invariants of knots are conceptually some of the simplest: given a knot K and a finite group G, the number of homomorphisms from the fundamental group of K to G is an invariant of K. Conceptual ease aside, the goal of this talk will be to explain my theorem with Greg Kuperberg that for nonabelian simple groups G, computing G-coloring invariants is #P-hard via almost parsimonious reduction. In particular, deciding if a knot K admits any nontrivial homomorphism to G is NP-hard. In analogy with famous results of Freedman-Larsen-Wang showing how to build a quantum computer with the Jones polynomial, our proof shows how to build a classical reversible computer using a combinatorial Aut(G)-equivariant TQFT associated to G.

Abstract: In the first part I will review some completion results for
topological spaces, and outline an approach to proving these using
higher Blakers-Massey type estimates, leading up to recent work of
Blomquist on iterated desuspensions of structured ring spectra (as
structured ring spectra). I the second part I will describe joint work
with Yu Zhang on constructing TQ-local homotopy theory and its
associated TQ-localization, together with recent results on homotopy
pro-nilpotent structured ring spectra. Finally, I will describe recent
joint work with Niko Schonsheck on TQ-completion for structured ring
spectra when certain connectivity assumptions are relaxed. Here, TQ is
short for topological Quillen homology, that is weakly equivalent to
stabilization.

(There will be an introductory talk)

Abstract:
In this talk I will show that two simply-connected spaces of finite type are rationally equivalent if and only if their singular cochains considered as associative algebras can be connected to each other by a zig-zag of quasi-isomorphisms of associative algebras.

This result is a consequence of the more general statement that two commutative algebras can be connected by a zig-zag of quasi-isomorphisms of commutative algebras if and only if they can be connected by a zig-zag of quasi-isomorphisms of associative algebras.

This is joint work with Ricardo Campos, Dan Petersen and Daniel Robert-Nicoud.

Abstract:
I will prove that in a stable range, the rational cohomology of the
moduli space of curves with level structures is the same as that of
the ordinary moduli space of curves: a polynomial ring in the MMM
classes. This can be viewed as a version of the Borel Stablity
Theorem for the moduli space of curves.

Abstract: We first give a basic introduction to Hochschild cohomology. Recall that the Hochschild cohomology of an algebra is defined as the Yoneda algebra of the identity bimodule in the derived category of bimodules. Hochschild cohomology has a rich algebraic structure: a Gerstenhaber algebra structure in cohomology and a B-infinity algebra structure at the cochain level.

Then we will talk about the singular Hochschild cohomology (or called Tate-Hochschild cohomology), which is defined as the Yoneda algebra of the identity bimodule in the singularity category of bimodules. The singularity category was introduced by Buchweitz in representation theory and then rediscovered by Orlov in algebraic geometry and homological mirror symmetry. We show that the singular Hochschild cohomology is endowed with the same rich structure as classical Hochschild cohomology. More precisely, we will construct an explicit complex to compute the Tate-Hochschild cohomology and show that there is a natural action of Kaufmann’s spineless cacti operad on this complex.

Abstract: In my talk, I will give an introduction to the group homology of the arithmetic groups SL_n Z and its prime level principal congruence subgroups. I will survey the known results and then report on progress on the Church-Farb-Putman Conjecture and the Lee-Szczarba Conjecture. This talk includes joint work with Jeremy Miller and Andrew Putman as well as work-in-progress with Jeremy Miller, Andrew Putman, and Jennifer Wilson.

Abstract: The factorization homology are invariants of n-dimensional manifolds with some fixed
tangential structures that take coefficients in suitable $E_n$-algebras. In this talk, I will give a definition for the equivariant factorization homology of a framed manifold for a finite group G via a monadic bar construction following Miller-Kupers. Then I will prove the equivariant nonabelian Poincare duality theorem in this case. As an application, in joint work with Asaf Horev and Inbar Klang, we calculate the equivariant factorization homology on equivariant spheres for certain Thom spectra.