Schedule

Abstract (Introduction to Foams I).  We explain the origins of foam theory in relation to link homology. Foam appears in the categorification of the Kuperberg sl(3) quantum invariant. 

Abstract (Introduction to Foams II).  This lecture will review the joint work with Louis-Hadrien Robert, circa 2018, on unoriented sl(3) foams and their evaluation.

Abstract.  These three lectures will be introductory, describing the basic constructions in gauge theory as applied to webs (trivalent graphs) embedded in 3-space, and foams (singular 2-complexes) embedded in 4-space. The talks are designed for non-specialists, with a particular focus on examples. 

The first lecture will focus primarily on webs in 3-space and their representation varieties, and cohomology classes thereon, with an emphasis on examples, before moving on to dimension four. The second lecture will take the four-dimensional constructions further, leading to the definition of the instanton evaluation of a foam together with some of the basic properties such as the neck-cutting relation. The third lecture will explore consequences of these relations, and will describe the limits of our present ability to calculate the instanton evaluation of a foam.

Abstract.  In this talk, I will present an evaluation formula for closed foams discovered with Louis-Hadrien Robert some years ago. This formula allows a simple definition of the gl(n) (exterior) Khovanov-Rozansky link homology, that I will recall. If time permits, I will give other applications: symmetric gl(n) KR link homologies, symmetries, etc. All results are joint with Louis-Hadrien Robert.

Abstract.  We discuss constructions of equivariant homology of links in the thickened annulus via foam evaluation. The thickened annulus is replaced by 3-space with a distinguished line. Foams may generically intersect the line, and they carry additional decorations at these intersection points. I will explain the construction in the sl(2) and sl(3) settings, based on joint work with Mikhail Khovanov, as well as in the gl(N) setting.

Abstract.  Cautis-Kamnitzer-Morrison discovered that SL_n webs can be studied through skew Howe duality. This approach was categorified by Lauda-Queffelec-Rose and Queffelec-Rose, giving rise to a generators and relations description of SL_n foams. I will discuss recent work with Daniel Tubbenhauer (arXiv:2303.04264) in which we study quantum skew Howe duality for symplectic groups. One notable outcome of our work is a canonical basis for the symplectic exterior algebra, which hints at the existence of a categorification. We explicitly describe the canonical bases and use them to study certain good filtrations related to modular representation theory. Time permitting, I will mention connections to webs for symplectic and orthogonal groups.

Abstract.  Kronheimer and Mrowka used gauge theory to define a functor J-sharp from a category of webs in R3 to the category of finite-dimensional vector spaces over the field of two elements. They also suggested a possible combinatorial replacement J-flat for J-sharp, which Khovanov and Robert proved is well-defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor J-sharp to the subcategory of planar webs is not the same as J-flat.

Abstract.  Using his instanton homology for closed 3-manifolds, Floer defined a knot invariant, which is known as instanton knot homology. Subsequently, Kronheimer and Mrowka proposed another approach to define knot invariants which are called reduced/unreduced singular instanton homology. In fact, the reduced version of singular instanton homology is isomorphic to instanton knot homology over the field of rational numbers. On the other hand, there is a direct relationship between instanton knot homology and Alexander polynomial whereas singular instanton homology is more directly related to Khovanov homology and Jones polynomial. For instance, there is an operator acting on instanton knot homology associated to any Seifert surface of the knot, which gives rise to a Z-grading on instanton knot homology. This Z-grading and the Floer grading can be used to identify instanton knot homology as a categorification of the Alexander grading of the knot. It is natural to ask whether one can define such operators on singular instanton knot homology. In this talk, I will explain how one can answer this question. In fact, one can define such operators on equivariant singular instanton homology, which has richer algebraic structures than singular instanton homology. Perhaps surprisingly, the extension of such operators to equivariant singular instanton homology are given by certain operators which have the form of analytical differential equations with singularities. This gives rise to certain additional structures on equivariant singular instanton homology that are formally similar to the doubly filtered knot Floer complex from Heegaard Floer homology. This talk is based on a work in progress with Chris Scaduto.

Abstract.  Various link homology theories associate a module over a polynomial ring in r (or more) variables to an r-component link. It turns out that this additional structure often helps to compute link homology, and the resulting modules are very interesting from the commutative algebra perspective. I will review these results for HOMFLY and Heegaard Floer homology, with (n,n) torus link as the main example. This is a joint work with Akram Alishahi and Beibei Liu.

Abstract.  I will explain the universal construction in topological theory and topological quantum field theory, with a focus on one-dimensional cobordisms with defects, and their relationship to pesudocharacters, an essential tool in modern number theory, and characters. If I have time, I will also discuss their connections to pseudo-holonomies and holonomies. This is joint with Mikhail Khovanov and Victor Ostrik.

Abstract.  I'll discuss some subtleties related to defining surgery sequences in instanton homology without using admissible bundles, by working through a specific example which demonstrates some of the danger. This is related both to joint work with Ali Daemi and Chris Scaduto, and to joint work with Tye Lidman and Josh Wang.

Abstract.  There is a categorification of the knot signature which is defined using singular instanton Floer theory with gauge group SU(2). I will discuss unoriented exact triangles for this theory and some resulting computations. A knot which has minimal rank Floer homology in this setting, equal to its Euler characteristic, will be called an "I-basic knot". A number of results suggest a strong connection between I-basic knots and L-space knots from Heegaard Floer theory. This is joint work with Ali Daemi.

Abstract.  Shumakovitch conjectured that the (unreduced) Khovanov homology of any nontrivial knot has 2-torsions. Inspired by the spectral sequence from Khovanov homology to singular instanton homology constructed by Kronheimer-Mrowka, we study the 2-torsions in unreduced variant of singular instanton homology for knots. When K is a fibered knot, we aim to show that the singular instanton homology has 2-torsions by comparing the homology groups with complex coefficients and Z/2 coefficients. Also, we aim to prove that the framed instanton homology of the closed 3-manifold obtained from S^3 by 1/2 surgery along any knot of genus > 1 has 2-torsions. This is a joint work in progress with Deeparaj Bhat and Zhenkun Li.