Homological Mirror Symmetry 2026

Maxim Kontsevich, IHES I

Tony Yue Yu, CalTech

Leonardo Cavenaghi, ICMS-Sofia: G-equivariant atoms (and symbols)

In this talk, we introduce the concept of G-equivariant Hodge atoms. We present many applications in G-equivariant birational geometry. We then explain how it can be merged with the theory of modular symbols, as developed by Kontsevich-Pestun-Tschinkel, and how to enhance the theory with data from Chen-Ruan cohomology. This is a joint work with L. Katzarkov and M. Kontsevich.

Maxim Kontsevich, IHES II

Tony Pantev, University of Pennslyvania

Semon Rezchikov, IHES

Refreshments

Umut Varolgunes, Koç University

Daniel Pomerleano, University of Massachusetts I

Paul Seidel, Massachusetts Institute of Technology

Maxim Kontsevich, IHES III

Daniel Halpern-Leistner, Cornell University

Andrew Harder, Lehigh University

Ron Donagi, University of Pennslyvania

Umut Varolgunes,  Koç University

Mohammed Abouzaid, Stanford University:  Generation Criteria

The standard generation criterion for Fukaya categories identifies when a finite collection of objects serve as split-generators. It has a more abstract formulation in terms of Calabi-Yau categories. I will discuss variants of this criterion that arise in different settings, e.g. when the category fails to be Calabi-Yau, or in filtered settings.

Daniel Pomerleano, University of Massachusetts II

Bernd Siebert, University of Texas at Austin: Pseudoholomorphic divisors in symplectic geometry

A fundamental difference in dealing with symplectic normal crossing divisors in symplectic versus complex geometry is the apparent absence of locally defining J-holomorphic functions. I will report on joint work with Yuan Yao showing that for any symplectic normal crossings divisor D there is nevertheless a path-connected space of tamed almost complex structures J with enough local J-holomorphic functions to describe the branches of D as zero loci. Using such adapted almost complex structures, the algebraic-geometric machinery of logarithmic geometry applies almost verbatim to the symplectic setting. A first application is a straightforward symplectic definition of logarithmic and punctured Gromov-Witten invariants.

Yan Soibelman, Kansas State University

David Favero, University of Minnesota

Shaoyun Bai, Massachusetts Institute of Technology: p-adic Gamma conjecture for toric Fano varieties

Following Seidel’s talk, we will discuss how to prove the p-adic Gamma conjecture for toric Fano varieties. The proof builds on pioneering work on p-adic cohomology of Dwork, Sperber and Adolphson-Sperber, and closed-string mirror symmetry also plays a crucial role.

Andrew Hanlon, University of Oregon:  The Cox category and homological mirror symmetry

The recently introduced Cox category is a natural repository for homological algebra on toric varieties and has a close relationship to homological mirror symmetry. This talk will focus on the relationship to HMS and raise open questions that may help generalize the construction.

Tobias Elkholm, Uppsala University:  Skein trace from curve counting

If M is a 3-manifold and L is a Lagrangian in the cotangent  bundle of M such that the projection of L to M is a branched cover  then there is a natural map from the skein of M to the skein of L.  Given a link in M, think of it as the boundary of a holomorphic curve  in the cotangent bundle and map it to the boundaries of all  holomorphic curves with boundary in L that has the given curve with  boundary in M as their thick part. When L is a double cover we obtain  explicit formulas for the lift by counting Morse flow trees. When M  and L are products of surfaces, our results give (skein lifts of)  Kontsevich-Soibelman wall crossing formulas, and (HOMFLYPT skein lifts  of) the Neitzke-Yan results for lifts of the gl(1) skein to the gl(2)  skein. The talk reports on joint work with Longhi, Park, and Shende.

Umut Varolgunes,  Koç University

Daniel Pomerleano, University of Massachusetts III

Jae Hee Lee, Stanford University

Amanda Hirschi, Sorbonne Université

Zihong Chen, Massachusetts Institute of Technology:  Kontsevich-Soibelman operations on the periodic cyclic homology

Deligne's conjecture (now a theorem) states that the Hochschild cochains of an associative algebra admits an action by the little 2-disks operad. This was generalized by Kontsevich and Soibelman, who constructed a 2-colored operad acting on the pair of Hochschild cochains and chains of a dg (or A_{\infty}) category. In this talk, I will answer the following questions: what equivariant homology operations does this 2-colored operad give rise to? It turns out the answer is only interesting in characteristic p, where these operations have a set of generators (under composition) previously studied in the context of equivariant Gromov-Witten theory. If time permits, I will talk about their connection to the Gauss-Manin connection, arithmetic aspects of Fukaya category, and symplectic topology. 

Sebastian Haney, Harvard University: Generalized holonomy and open Gromov-Witten invariants

The open Gromov-Witten potential of a Lagrangian submanifold L of a symplectic manifold is a function from the space of bounding cochains on L, whose values can be interpreted as counts of holomorphic disks with boundary on L. I will describe a construction of the open Gromov-Witten potential which realizes it as a class in the cyclic cohomology of the Fukaya A-infinity algebra of L. As a byproduct of this construction, one obtains integer-valued open Gromov-Witten invariants whenever the Floer cohomology of L is defined over the integers. I will also explain how these results may be used to compare various definitions of the open Gromov-Witten invariants, their relation to the cohomology of the free loop space, and some topological consequences of the (non-)integrality of disk counts.