Wednesday

We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.

For the past several decades, SU(2) instanton Floer theory has provided numerous applications to low-dimensional topology, especially to problems involving fundamental groups of 3-manifolds. While versions of instanton Floer homology have been constructed for higher rank Lie groups such as SU(N) for any N, these variants of the theory have not yet been used to prove new topological results. I will explain in this talk the first such applications in the case N = 3. I will also discuss a structure theorem for SU(3) Donaldson-type invariants of 4-manifolds, among other topics. Based on joint work with Ali Daemi and Nobuo Iida.

Z2-Harmonic 1-forms are singular generalizations of classical harmonic 1-forms that allow topological twisting around a link in a 3-manifold. Such objects are expected to provide a gauge-theoretic refinement of the classical Morgan-Shalen compactification of the SL(2,C) character variety, and work of Taubes and Witten suggests this connection intertwines these objects with several distinct areas of classical 3-manifold topology. In this talk, I will present some new techniques for constructing Z2-harmonic 1-forms via gauge-theoretic methods, and discuss recent gluing results in Seiberg-Witten theory that provide a template for constructing the compactification. Parts of this talk are based on joint work with Siqi He.

In this talk we illustrate the most salient features of boundary value problems for non-abelian gauge theories over Riemannian manifolds with boundary, and their applications and implications in connection to the fields of geometric analysis and mathematical physics. We define a range of well-posed boundary value problems for the Yang-Mills/ Yang-Mills-Higgs equations, analyzed in comparison to other contexts, such as that of harmonic maps, H-surface equations, and, possibly, the nonlinear sigma model. A Morse theory for the Yang-Mills Dirichlet problem is outlined. This talk includes results derived by the author, individually and collaboratively, over the course of many years [with Isobe, on the existence of non-absolute minimizers and min-max-type solutions, with Moncrief and Maitra, for applications to QFT, with Otway, to frame those results in the context of a nonlinear Hodge-de Rham theory].

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.