Workshop

Talk 1: Julia Semikina - "Cut-and-Paste Invariants of Manifolds via K-Theory (Part 1)"

In this talk I will explain the construction due to Campbell and Zakharevich and its application that will allow us to speak about the "K-theory of manifolds" spectrum. The K_0 of the constructed spectrum recovers the (almost) classical SK-groups introduced by Kreck, Karras, Neumann and Ossa. I will explain how to relate the spectrum to the algebraic K-theory of integers, and how this leads to certain classical invariants of manifolds when restricted to the lower homotopy groups.

Talk 2: Renee Hoekzema - "Cut-and-Paste Invariants of Manifolds via K-Theory (Part 2)"

Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory. In this talk I present a new application of their framework to the study of cut-and-paste invariants of manifolds. We construct a spectrum that recovers the classical SK (“schneiden und kleben,” German for “cut and paste”) groups for manifolds on the zeroth homotopy group, and we construct a lift of the Euler characteristic, one of these invariants, to a map of spectra. This is joint work with M. Merling, L. Murray, C. Rovi and J. Semikina.

Talk 3: Ipsita Datta - "Obstructions to the Existence of Lagrangian Cobordisms"

We present new obstructions to the existence of Lagrangian cobordisms in $\mathbb{R}^4$ which are developed by studying holomorphic curves. We study related object called Lagrangian tangles so that we don't need to assume that the boundaries of the Lagrangians are Legendrian in any contact structure.

Talk 4: Orsola Capovilla-Searle - "Obstructions to Reversing Lagrangian Surgery"

Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

Talk 5: Weizhe Shen - "Rank Inequalities for the Knot Floer Homology of (1,1)-Satellites"

One application of the immersed-curve technique, introduced by Hanselman-Rasmussen-Watson, is to study rank inequalities for Heegaard Floer homology in the presence of certain degree-one maps. Another application, discovered by Chen, is to describe the knot Floer homology of satellite knots with (1,1)-patterns. We will discuss similar rank inequalities for the knot Floer homology of (1,1)-satellites based on Chen's work.

Talk 6: Catherine Cannizzo - "Floer Theory of a Symplectic Fibration"

Lagrangian intersection Floer theory defines morphisms in the Fukaya category of a closed symplectic manifold, or the Fukaya-Seidel category of a symplectic fibration over a two dimensional base with boundary. We give an example of this cohomology theory in a certain symplectic fibration with T^4 fibers, critical locus given by the banana manifold, and "U-shaped" Lagrangians.

In this talk, I will report recent results about quantizations of exact Lagrangians using microlocal sheaves of spectra, and some applications. These include a complete characterization of obstructions to quantizations which can be seen as an upgrade of the Maslov index to the higher topological levels, and an alternative proof of a result of Abouzaid--Kragh concerning compact exact smooth Lagrangians in cotangent bundles.

Factorization homology arose from Beilinson-Drinfeld's algebro-geometric approach to conformal field theory, and from study of labeled configuration spaces due to McDuff, Segal, Salvatore, Andrade, and others. In this talk, I will give an introduction to factorization homology and equivariant factorization homology from the point of view of configuration spaces. I will then discuss joint work with Asaf Horev and Foling Zou, with an appendix by Jeremy Hahn and Dylan Wilson, in which we prove a "non-abelian Poincaré duality" theorem for equivariant factorization homology, and study the equivariant factorization homology of equivariant Thom spectra. In particular, this provides an avenue for computing certain equivariant analogues of topological Hochschild homology.