# Floer homology, low-dimensional topology, and algebra

# Floer homology, low-dimensional topology, and algebra

**9:30 AM — 10:30 AM****Speaker: Boyu Zhang, ****University of Maryland at College Park****Location: McDonnell A02**

*Ring structures on singular instanton homology and link detections*

The singular instanton Floer homology of S^1 times a surface has a natural ring structure given by the pair-of-pants cobordisms. The study of this ring structure and related questions has a long history that can be dated back to the work of Atiyah and Bott. In this talk, I will present a complete characterization of the ring structure on singular instanton homology in C coefficients. I will then present several applications of this computation in the study of knots and links. For example, we show that if L is a link in S^3 that is not the unknot or the Hopf link, then the fundamental group of the complement of L has an irreducible SU(2) representation; we also give a complete classification for links whose Khovanov homology have the minimal possible rank.

**10:45 AM — 11:45 AM****Speaker: İnanç Baykur, ****University of Massachusetts Amherst****Location: McDonnell A02**

*A dichotomy in dimension four*

Does every four-manifold admit either no smooth structure or infinitely many of them? I'll report on recent work, joint with A. Stipsicz and Z. Szabó, addressing this question for four-manifolds with finite cyclic fundamental groups. A bonus discussion may feature fake projective planes in the context of yet another dichotomy.

**1:30 PM — 2:30 PM****Speaker: Sherry Gong, ****Texas A&M University****Location: McDonnell A01**

*Ribbon concordances and slice obstructions: experiments and examples*

We will discuss some computations of ribbon concordances between knots and talk about the methods and the results. This is a joint work with Nathan Dunfield.

**3:00 PM — 4:00 PM****Speaker: András Juhász, ****University of Oxford****Location: McDonnell A01**

*Examples of topologically unknotted tori*

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.