Homological Mirror Symmetry 2026

Maxim Kontsevich, IHES I: On theory of atoms

I'll recall the basics of theory of atoms, its applications to rationality questions and to stability structures. I'll also discuss some issues with fields of definition, and an extension of theory to the relative case.

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.

Ron Donagi, University of Pennslyvania: Some Classification Questions: Algebraic Geometry vs. SQFT

Seiberg and Witten showed that some supersymmetric quantum field theories can be solved mathematically. Subsequent work suggested that every N=2 susy theory in 4D leads to an algebraically completely integrable system, which can be solved in terms of theta functions. The base of this system is the Coulomb branch of the theory, a vector space whose dimension is called the rank. The theory has a symmetry governed by the ‘flavor’ Lie algebra. A heroic series of works of Argyres et al gave a complete classification of the resulting integrable systems in rank 1. (Depending on exactly what you count, they get 35 or 28 or 27 such systems.)

In equally heroic works on the algebraic geometry side, Persson and Miranda classified all 279 types of rational elliptic surfaces in terms of their configurations of singular fibers. Subsequent work of Oguiso-Shioda determined the corresponding Mordell-Weil groups of sections, and Karayayla’s thesis determined the automorphism groups.

In this talk we will explore the math/physics dictionary. The integrable systems arising from SQFT live on rational elliptic surfaces. Imposing some very simple constraints reduces the Persson-Miranda list to the physicists’ list. Some natural expansions of this list arise from Karayayla’s work. The flavor symmetry is interpreted in terms of the Mordell-Weil group. Many questions remain open.

Maxim Kontsevich, IHES II: More on non-holomorphic Landau-Ginzburg model

This lecture is a continuation of my talks in the Miami conference in 2023, and will serve as a motivation for the hypothetical picture of moving spectra in the talk by T.Pantev.

Tony Pantev, University of Pennslyvania: Integral structures on Hodge atoms

I will describe a novel formalism for defining and specifying  integral structures on non-archimedean F-bundles and on atoms. I will explain a construction of the Gamma-corrected integral structure on the A-model non-archimedean F-bundle and will discuss the special features of the Gamma-conjecture that are important in  this setting. This is a report on a joint work in progress with Katzarkov, Kontsevich and Yu.

Semon Rezchikov, IHES: Noncommutative Cartier Formulae

I will discuss some of the details of proof of the computation of the p-curvature of the Getzler connection on a family of categories, relation to the behavior of noncommutative calculus in positive characteristic. Given a geometric analog of the cyclotomic structure map (which is still to be made precise), an analogous argument would compute the p-curvature of the small connection for an arbitrary symplectic Calabi-Yau.

Refreshments

Umut Varolgunes, Koç University I: Homological mirror symmetry via Floer theory with support

Let X be a geometrically bounded and graded symplectic manifold equipped with a continuous involutive map X\to B - a Maslov 0 Lagrangian torus fibration with singularities admitting a Lagrangian section L and satisfying various other properties. A long standing project with Abouzaid and Groman aims to give a Floer theoretic construction of a smooth non-archimedean analytic CY space Y over \Lambda=k((T^\bR)), where k is trivially valued, so that there is a manifest cohomologically fully faithful functor Fuk(M, \Lambda)\to Coh^dg(Y). If X is a Liouville manifold we can also work over k. The construction relies on Fukaya categories with support on compact subsets and is inspired by Kontsevich-Soibelman's study of symplectic K3's. Given a decomposition of B into convex rational polytopes, one expects to build a Reynaud model of Y. Going one step further, jointly with Gao, we observe that if B is equipped with a Gross-Siebert toric polyhedral decomposition (B,P) and X with a Seidel large volume limit (X,D,\theta), compatible with each other and L, one should be able to construct a formal scheme \cY over DVR k[[Q]] and a fully faithful functor from a full subcategory of Fuk(X,D) to Perf^dg(\cY). This construction relies on Sheridan's relative Fukaya category with support on the pieces of the decomposition of X. If (B,P) admits a polarization \varphi (which gives rise to a monodromy symplectomorphism and a good supply of Lagrangians in X), we can show that \cY is projective (with a homogeneous coordinate ring constructed from the Floer theory of L and its iterated monodromy images in the expected way) and the image of the functor split-generates. We also expect that \cY recovers Gross-Siebert's mirror toric degeneration to X from their Annals paper. Assuming that the base change of \cY to \Lambda is smooth, this base change should give an algebraic model of Y from the AGV construction, and (independently) we obtain an HMS equivalence in original form using automatic generation of Ganatra-Sanda. Finally, if the induced decomposition of X-D has pair-of-pants pieces, we claim that the subcategory actually split-generates Fuk(X,D), giving rise to an equivalence between the split-triangulated closure of Fuk(X,D) and Perf^dg(\cY).

In the first lecture I will give an overview and then focus on the mirror space construction with comparisons to literature. In the second one I will focus on the constructions of the various Fukaya categories with support and the HMS functors. In the last one I will discuss examples, sticking mostly to CY hypersurfaces in smooth toric Fanos.

Daniel Pomerleano, University of Massachusetts I: QH^* of GIT quotients, shift operators, and SH^* with supports

The quantum cohomology of GIT quotients is a classical subject that has been extensively studied from diverse perspectives. My lectures will focus on a recent framework (due to Teleman) which provides a formula for the quantum cohomology rings of GIT quotients X//G in terms of non-abelian ``shift operators". I hope to emphasize the role of Varolgunes' "symplectic cohomology with supports" in this setting, particularly as a tool for extending these results beyond the monotone setting.

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.

Maxim Kontsevich, IHES III: Reidemeister Trace for Frobenius automorphism

By classical Nielsen-Reidemeister theory, with any homotopy type of a self-map of a finite CW complex one can associate a trace which takes values in the 0-th Hochschild homology of the group rings of the fundamental group of the mapping cylinder. Traces of the iterations of the self-map can be organized in a kind of non commutative rational function. Conjecturally, a similar rationality holds for the Frobenius automorphism for any variety over a finite field.  

Daniel Halpern-Leistner, Cornell University: Polyhedral decompositions of equivariant derived categories

I will discuss a new family of semiorthogonal decompositions for equivariant derived categories of coherent sheaves. It combines methods developed by Špenko and van den Bergh with earlier work by me and Ballard-Favero-Katzarkov. These semiorthogonal decompositions allow one to compare the derived categories of different GIT quotients, and suggest an approach to studying the D-equivalence conjecture in this context that is closely related to the noncommutative minimal model program.

Andrew Harder, Lehigh University: Towards mirror P=W for Clarke pairs 

Amanda Hirschi, Sorbonne Université: An open-closed Deligne-Mumford field theory associated to a Lagrangian

In 2007, Costello outlined a programme to show that homological mirror symmetry implies enumerative symmetry, using the notion of an open-closed topological conformal field theory (TCFT). I will describe the construction of an open-closed DMFT, a variant of an open-closed TCFT, from moduli spaces of stable pseudo-holomorphic curves with boundary on a single embedded Lagrangian in a closed symplectic manifold. This is joint work with Kai Hugtenburg.

Umut Varolgunes, Koç University II: Homological mirror symmetry via Floer theory with support

Let X be a geometrically bounded and graded symplectic manifold equipped with a continuous involutive map X\to B - a Maslov 0 Lagrangian torus fibration with singularities admitting a Lagrangian section L and satisfying various other properties. A long standing project with Abouzaid and Groman aims to give a Floer theoretic construction of a smooth non-archimedean analytic CY space Y over \Lambda=k((T^\bR)), where k is trivially valued, so that there is a manifest cohomologically fully faithful functor Fuk(M, \Lambda)\to Coh^dg(Y). If X is a Liouville manifold we can also work over k. The construction relies on Fukaya categories with support on compact subsets and is inspired by Kontsevich-Soibelman's study of symplectic K3's. Given a decomposition of B into convex rational polytopes, one expects to build a Reynaud model of Y. Going one step further, jointly with Gao, we observe that if B is equipped with a Gross-Siebert toric polyhedral decomposition (B,P) and X with a Seidel large volume limit (X,D,\theta), compatible with each other and L, one should be able to construct a formal scheme \cY over DVR k[[Q]] and a fully faithful functor from a full subcategory of Fuk(X,D) to Perf^dg(\cY). This construction relies on Sheridan's relative Fukaya category with support on the pieces of the decomposition of X. If (B,P) admits a polarization \varphi (which gives rise to a monodromy symplectomorphism and a good supply of Lagrangians in X), we can show that \cY is projective (with a homogeneous coordinate ring constructed from the Floer theory of L and its iterated monodromy images in the expected way) and the image of the functor split-generates. We also expect that \cY recovers Gross-Siebert's mirror toric degeneration to X from their Annals paper. Assuming that the base change of \cY to \Lambda is smooth, this base change should give an algebraic model of Y from the AGV construction, and (independently) we obtain an HMS equivalence in original form using automatic generation of Ganatra-Sanda. Finally, if the induced decomposition of X-D has pair-of-pants pieces, we claim that the subcategory actually split-generates Fuk(X,D), giving rise to an equivalence between the split-triangulated closure of Fuk(X,D) and Perf^dg(\cY).

In the first lecture I will give an overview and then focus on the mirror space construction with comparisons to literature. In the second one I will focus on the constructions of the various Fukaya categories with support and the HMS functors. In the last one I will discuss examples, sticking mostly to CY hypersurfaces in smooth toric Fanos.

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: QH^* of GIT quotients, shift operators, and SH^* with supports

The quantum cohomology of GIT quotients is a classical subject that has been extensively studied from diverse perspectives. My lectures will focus on a recent framework (due to Teleman) which provides a formula for the quantum cohomology rings of GIT quotients X//G in terms of non-abelian ``shift operators". I hope to emphasize the role of Varolgunes' "symplectic cohomology with supports" in this setting, particularly as a tool for extending these results beyond the monotone setting.

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: Geometry of the generalized local Riemann-Hilbert correspondence.

Generalized Riemann-Hilbert correspondence (GRHC) was formulated by Maxim Kontsevich and myself more than 10 years ago as a part of our program "Holomorphic Floer Theory". The GRHC relates deformation quantization of complex symplectic manifolds with their Floer theory. Few years ago we formulated a local version of GRHC in which the relevant categories are associated with a fixed complex analytic Lagrangian subset of a complex symplectic manifold. In my talk I am going to discuss  new phenomena which we discovered in the course of study of local GRHC. One of them is a generalization of the notion of support of an object of the Fukaya category. Another one is a generalization of Orlov theorem about the category of perfect complexes on a blow-up to the framework of deformation quantization.

David Favero, University of Minnesota: Locally-Free Resolutions on toric varieties

Hochster's Formula gives a description of the Betti numbers of a monomial ideal in terms of the cohomology of an associated simplicial complex.  I will discuss a toric generalization of Hochster's Formula based on joint work with M. Sapronov.    Roughly, the strategy is to view the associated simplicial complex as the homologically mirror Lagrangian to the module. 

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.

Shaoyun Bai, Massachusetts Institute of Technology: Overconvergent Frobenius intertwiner for quantum connections and p-adic Gamma class

Differential equation over p-adic fields is one of the most interesting aspects of arithmetic geometry. Inspired by this, it is tempting to study the quantum connections base-changed to p-adic fields. In this talk, I will explain the notion of Frobenius intertwiner and its convergence properties, in order to formulate a conjecture involving extraordinary convergence properties of Frobenius intertwiners of quantum connections of Fano varieties. The p-adic analog of the Gamma function plays a crucial role here. Time permitting, I will discuss how to prove the conjecture for toric Fano varieties, which combines mirror symmetry and pioneering work of Dwork, Sperber, and Adolphson. This is based on joint work with Pomerleano and Seidel.

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.

Umut Varolgunes, Koç University III: Homological mirror symmetry via Floer theory with support

Let X be a geometrically bounded and graded symplectic manifold equipped with a continuous involutive map X\to B - a Maslov 0 Lagrangian torus fibration with singularities admitting a Lagrangian section L and satisfying various other properties. A long standing project with Abouzaid and Groman aims to give a Floer theoretic construction of a smooth non-archimedean analytic CY space Y over \Lambda=k((T^\bR)), where k is trivially valued, so that there is a manifest cohomologically fully faithful functor Fuk(M, \Lambda)\to Coh^dg(Y). If X is a Liouville manifold we can also work over k. The construction relies on Fukaya categories with support on compact subsets and is inspired by Kontsevich-Soibelman's study of symplectic K3's. Given a decomposition of B into convex rational polytopes, one expects to build a Reynaud model of Y. Going one step further, jointly with Gao, we observe that if B is equipped with a Gross-Siebert toric polyhedral decomposition (B,P) and X with a Seidel large volume limit (X,D,\theta), compatible with each other and L, one should be able to construct a formal scheme \cY over DVR k[[Q]] and a fully faithful functor from a full subcategory of Fuk(X,D) to Perf^dg(\cY). This construction relies on Sheridan's relative Fukaya category with support on the pieces of the decomposition of X. If (B,P) admits a polarization \varphi (which gives rise to a monodromy symplectomorphism and a good supply of Lagrangians in X), we can show that \cY is projective (with a homogeneous coordinate ring constructed from the Floer theory of L and its iterated monodromy images in the expected way) and the image of the functor split-generates. We also expect that \cY recovers Gross-Siebert's mirror toric degeneration to X from their Annals paper. Assuming that the base change of \cY to \Lambda is smooth, this base change should give an algebraic model of Y from the AGV construction, and (independently) we obtain an HMS equivalence in original form using automatic generation of Ganatra-Sanda. Finally, if the induced decomposition of X-D has pair-of-pants pieces, we claim that the subcategory actually split-generates Fuk(X,D), giving rise to an equivalence between the split-triangulated closure of Fuk(X,D) and Perf^dg(\cY).

In the first lecture I will give an overview and then focus on the mirror space construction with comparisons to literature. In the second one I will focus on the constructions of the various Fukaya categories with support and the HMS functors. In the last one I will discuss examples, sticking mostly to CY hypersurfaces in smooth toric Fanos.

Daniel Pomerleano, University of Massachusetts III: QH^* of GIT quotients, shift operators, and SH^* with supports

The quantum cohomology of GIT quotients is a classical subject that has been extensively studied from diverse perspectives. My lectures will focus on a recent framework (due to Teleman) which provides a formula for the quantum cohomology rings of GIT quotients X//G in terms of non-abelian ``shift operators". I hope to emphasize the role of Varolgunes' "symplectic cohomology with supports" in this setting, particularly as a tool for extending these results beyond the monotone setting.

Jae Hee Lee, Stanford University: Quantum power operations and p-curvature

The quantum connection can be defined integrally for sufficiently positive symplectic manifolds, allowing one to consider their characteristic p reductions. We identify such mod p quantum connections with the action of quantum deformations of Steenrod operations in many cases, notably for all Calabi--Yau threefolds. The story also admits a q-deformation, involving Adams operations in K-theory. Based on joint work with Shaoyun Bai.

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. 

Tony Yue Yu, CalTech: Decomposition of F-bundles

I will discuss various aspects of decomposition of F-bundles: motivations from homological mirror symmetry, the spectral decomposition theorem, uniqueness of decomposition, decomposition for projective bundles, blowups, variations of GIT quotients and local models of standard flips in binational geometry. Based on arXiv preprints 2411.02266, 2505.09950, 2508.05105, 2508.15770.