Federico Cantero Morán, "Higher Steenrod squares for Khovanov homology"
Abstract: In 2012 Lipshitz and Sarkar associated to each link a family of spectra that refined Khovanov homology. As a consequence, Khovanov homology with F_2-coefficients became endowed with Steenrod squares. Shortly after, they gave a formula to compute the second Steenrod square, and showed that it is often non-trivial. In this talk we will give explicit formulas, using cup-i products, to compute the higher Steenrod squares. (https://arxiv.org/abs/1902.02839)
Irving Dai, Matthew Stoffregen, and Linh Truong, "An infinite rank summand of the homology cobordism group"
Abstract: We show that the homology cobordism group of integer homology three-spheres contains an infinite rank summand. The proof uses an algebraic modification of the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is inspired by Hom’s techniques in the setting of knot concordance. This is joint work with Irving Dai, Jen Hom, Matt Stroffregen, and Linh Truong.
Hokuto Konno, "The diffeomorphism and homeomorphism groups of 4-manifolds"
Abstract: I will explain a family version of 10/8-type inequalities. Although, for now, this inequality is shown only for limited spin 4-manifolds, that extracts a homotopical difference between the diffeomorphism and homeomorphism groups. More precisely, I will show that, for some 4-manifolds, the inclusions from the diffeomorphism groups into the homeomorphism groups are not weak homotopy equivalences. It is interesting to compare this result with manifolds of lower dimension: for a closed manifold of dimension less than 4, the inclusion from the diffeomorphism group into the homeomorphism group is known to be a weak homotopy equivalence. To detect the difference in dimension 4, we will construct non-smoothable topological bundles with smooth 4-manifolds fibers. This talk is based on joint work with Tsuyoshi Kato and Nobuhiro Nakamura.
Tim Large, "Localization and Steenrod operations"
Abstract: Following the guidance of Kronheimer and Mrowka’s approach to monopole Floer homology, we develop a model for Z/2-equivariant symplectic Floer theory using equivariant almost complex structures. The resulting equivariant Floer cohomology admits a localization map to a twisted version of Floer cohomology in the invariant set of the Z/2-action, generalising the work of Seidel and Smith to settings where the normal bundle to the invariant set is not stably trivial. We then present two applications. First, we obtain a definition of Steenrod squares on F_2-coefficient Floer cohomology, without an appeal to stable homotopy types. Second, we study the Heegaard Floer homology of double-covers of three-manifolds, extending the work of Hendricks and Lipshitz-Treumann.
Jianfeng Lin, XiaoLin Danny Shi, and Zhouli Xu, "The geography problem of 4-manifolds: 10/8+4"
Abstract: A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.
Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to Furuta's problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This talk is based on joint work by Mike Hopkins, Jianfeng Lin, XiaoLin Danny Shi and Zhouli Xu.
S. Michael Miller, "Equivariant instanton homology and group cohomology"
Abstract: Floer's celebrated instanton homology groups are defined for integer homology spheres, but analagous groups in Heegaard Floer and Monopole Floer homology theories are defined for all 3-manifolds; these latter groups furthermore come in four flavors, and carry extra algebraic structure. Any attempt to extend instanton homology to a larger class of 3-manifolds must be somehow equivariant - respecting a certain SO(3)-action. We explain how ideas from group cohomology and equivariant algebraic topology allow us to define four flavors of instanton homology for rational homology spheres, and how these invariants relate to existing instanton homology theories.
Patrick Orson, "Calculations with flow categories"
Abstract: I will discuss the approach to the Khovanov stable homotopy type using the flow category technology of Cohen-Jones-Segal, which is how Lipshitz and Sarkar originally constructed these spaces. One advantage of this technology is that it can be used in a situation that is not necessarily modelled on a cubical diagram. I will describe work with Andrew Lobb and Dirk Schuetz to develop a useful computational calculus for flow categories, which can help in the calculation of knot homotopy types “by hand”.
Ian Zemke, "Doubling tricks and knot Floer homology"
Abstract: Given a concordance C from K0 to K, one can form a concordance from K0 to K0 by doubling C. A Morse function on C gives a simple description of the doubled concordance in terms of adding tubes and tubing on spheres, which in turn gives algebraic restrictions on the knot Floer homologies of K0 and K. One application of this perspective is a simple proof that the map on knot Floer homology induced by a ribbon concordance is injective. Another application is that the U torsion order of knot Floer homology gives a lower bound on the bridge index of a knot. Some of this work is joint (in various pieces) with Adam Levine, Maggie Miller, and Andras Juhasz.
Melissa Zhang, "Localization in Khovanov homology"
Abstract: We use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov stable homotopy type to prove localization theorems and Smith-type inequalities for the Khovanov homology of periodic links. This is joint work with Matthew Stoffregen.
Short talks