• Title: Towards extended Floer field theories
• Date: 02/01/2018
• Time: 2:00 PM - 2:50 PM
• Place: C304 Wells Hall
• Speaker: Guillem Cazassus, Indiana University

Donaldson polynomials are powerful invariants associated to smooth four-manifolds. The introduction by Floer of Instanton homology groups, associated to some 3-manifolds, allowed to define analogs of such polynomials for (some) four-manifolds with boundary, that have a structure similar with a TQFT. Wehrheim and Woodward developed a framework called "Floer field theory" which, according to the Atiyah-Floer conjecture, should permit to recover Donaldson invariants from a 2-functor from the 2-category Cob_{2+1+1} to a 2-category Symp they defined, which is an enrichment of Weinstein's symplectic category. I will describe a framework that should permit to extend such a 2-functor to lower dimensions. This framework should permit to define new invariants in Manolescu and Woodward's symplectic instanton homology (sutured theory, equivariant version). This is work in progress.

• Title: Triple linking numbers and surface systems
• Date: 03/01/2018
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Matthias Nagel, McMaster University, Canada

We relate fillability of two link exteriors, and the question when two links admit homeomorphic surface systems to (a refinement of) Milnor’s triple linking numbers. This extends a theorem of Davis-Roth to include also links with non-vanishing linking numbers. This is joint work with C. Davis, P. Orson, and M. Powell.

• Title: Connected Heegaard Floer homology and homology cobordism
• Date: 03/15/2018
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Kristen Hendricks, MSU

We study applications of Heegaard Floer homology to homology cobordism. In particular, to a homology sphere Y, we define a module HF_conn(Y), called the connected Heegaard Floer homology of Y, and show that this module is invariant under homology cobordism and isomorphic to a summand of HF_red(Y). The definition of this invariant relies on involutive Heegaard Floer homology. We use this to define a new filtration on the homology cobordism group, and to give a reproof of Furuta's theorem. This is joint work with Jen Hom and Tye Lidman.

• Title: Asymptotic of quantum representations of surface groups
• Date: 03/29/2018
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Ramanujan Santharoubane, University of Virginia

In a previous work with Thomas Koberda we defined actions of surface groups on the vector spaces coming from the Witten-Reshetikhin-Turaev TQFT. For l a given loop in a surface we can define the trace of the associated operator. Actually, this is a sequence of invariant depending on a sequence of roots of unity. For any z on the unit circle, we study the asymptotic of this sequence of invariant when the sequence of roots of unity converges to z. The main theorem says that this asymptotic is determined by the evaluation at z of a Laurent polynomial depending only on l. This polynomial can be viewed as a Jones polynomial for surface groups. The main corollary concerns the so-called AMU conjecture which relates TQFT representations of mapping class groups to the Nielsen-Thurston classification. This talk represent a joint work with Julien Marché.

• Title: Seiberg-Witten monopoles with multiple spinors on a surface times a circle
• Date: 04/10/2018
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Aleksander Doan, Stony Brook University

I will discuss a generalisation of the 3-dimensional Seiberg-Witten equations which was studied by Haydys and Walpuski in relation to Yang-Mills theory on manifolds with special holonomy. The main difference from the classical setting is the non-compactness of the moduli space of solutions. I will explain how to tackle this problem and count the solutions in the special case when the underlying 3-manifold is the product of a Riemann surface and a circle. The main ingredient is a holomorphic description of the moduli space of solutions and its compactification. It allows us to relate our problem to classical results on holomorphic vector bundles on Riemann surfaces.

• Title: Topological Hochschild homology and logarithmic geometry
• Date: 04/19/2018
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Speaker: Calvin Woo, Indiana University

While a need to compute algebraic K-theory led topologists to consider spectrally-enriched versions of Hochschild homology, over the years topological Hochschild homology (THH) has emerged as an interesting invariant in its own right. In this talk, I will introduce some of these interesting properties and show how logarithmic geometry can help us shine light on THH's arithmetic structure.