2022 Simons Collaboration on Homological Mirror Symmetry Annual Meeting
Organizer:
Tony Pantev, University of Pennsylvania
Previous meeting pages:

Meeting Report
The meeting highlighted major advances and applications obtained by the methodology developed by the collaboration in the past seven years. Aside from a comprehensive overview talk by Abouzaid all the talks reported on landmark new results by former and current graduate students and postdocs in the collaboration.
• Abouzaid surveyed the key areas of symplectic topology, noncommutative geometry and quantum field theory that underwent extensive developments through the works of the collaboration members. Landmarks here included the proof of mirror symmetry without corrections and near immersed Lagrangians, local to global approaches to constructing mirrors, the categorical reconstruction of Hodge theoretic mirror symmetry, the newly discovered enumerative meaning of the arithmetic of periods, noncommutative gluing theorems, the proof of mirror symmetry for singular spaces, and the integrality of noncommutative Hodge structures of symplectic The emphasis was on the resolution of longstanding conjectures, the introduction of new perspective on important problems and the opening of completely unexpected new connections.
• In his talk, Hicks explained how the classical Floer theory of monotone Lagrangian cobordisms allows us to describe iterated mapping cones between general objects of the Fukaya category. The construction utilizes generators in the Fukaya category and exhibits a Lagrangian cobordism from the generating Lagrangians to the desired general Hicks discussed strong applications in HMS where the monotonicity requirement is met, analyzed the unobstructedness beyond the monotone setting and demonstrated how the geometry of Lagrangian cobordisms determines unobstructedness. Sample computations for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds were also presented.
• In her lecture, Ward described her recent joint work with Keating which completes a major step in the HMS Collaboration’s program of extracting new birational invariants of algebraic varieties from symplectic topology. The exciting new direction initiated by Keating and Ward gives a universal description of the symplectomorphisms mirror to the elements in the Cremona group. In particular, Keating and Ward construct a universal fourdimensional Weinstein domain of infinite type and prove an HMS correspondence between distinguished birational transformations of the projective plane preserving a standard holomorphic volume form and symplectomorphisms of this universal domain. The Keating–Ward universal domain contains every Liouville manifold mirror to a log Calabi–Yau surface as a Weinstein subdomain. In particular, Viterbo restriction to these subdomains recovers a mirror correspondence between the automorphism group of any open log Calabi–Yau surface and the symplectomorphism group of its mirror. This exciting progress also leads to a precise conjectural HMS correspondence between the full group of volumepreserving birational transformations of the plane and a group of symplectomorphisms of nonexact symplectic deformations of the Keating–Ward domain.
• Yu reported on a breakthrough progress, joint with Keel, on proving positivity and integrality of the mirror structure constants for a log Calabi–Yau variety. In particular, Yu presented a new construction of the structure constants of the mirror algebra of theta functions for a smooth affine log Calabi–Yau variety. The construction utilizes delicate counts of nonArchimedean analytic disks in the skeleton of the Berkovich analytification of the Calabi–Yau variety. This generalizes previous constructions of Gross–Hacking–Keel and Keel–Yu which relied on extra toric assumptions. The new proof is based on an analytic modification of the target space as well as the theory of skeletal curves. The toolbox Keel–Yu developed for carrying out this proof has a much wider range of applicability, and several generalizations and comparisons of virtual fundamental classes were discussed.
• In her lecture, Argüz described exciting new progress understanding the relationship between Gross–Siebert intrinsic mirrors and Fock–Goncharov dual cluster varieties. A sequence of recent works of Argüz, Argüz–Gross, and Argüz–Bousseau bridge the theory of cluster varieties with the algebrogeometric framework of Gross–Siebert mirror symmetry. The precise result is that the mirror to a cluster variety is a degeneration of the Fock–Goncharov dual cluster The proof and the algorithmic construction of the requisite degenerations are based on a delicate comparison of the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich and the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions.
• Yuan reviewed a powerful new construction of the mirror space of a local SYZ singularity. The construction utilizes the family Floer theory developed by Abouzaid, Fukaya and other members of the collaboration in the past seven years. The mirror rigid analytic fibration (including the singular fibers) arises as a modification of the singular nonArchimedean model of Kontsevich–Soibelman and also agrees with the construction of Gross–Hacking–Keel–Siebert. In the process, Yuan identifies the conceptual obstruction that prevented the discovery of this construction for many years. Surprisingly, the nonArchimedean topology requires a modification of the conventional Maurer–Cartan picture since the dual singular fiber can be larger than the Maurer–Cartan set. The dual singular fiber also has an intrinsic geometric description relying on the classification of two types of quantum corrections and the theory of Chambert–Loir and Ducros. The new approach leads to the first unconditional mathematically precise Tduality interpretation of mirror symmetry and provides string supporting evidence for a cornerstone conjecture on the behavior of Bmodel noncommutative Hodge structures.
• Wilkins described his new method of equivariant localization construction of Borman–Sheridan classes. The method is based on a surprising, upgraded Atiyah–Bott localization mechanism that can be applied to pseudocycle bordisms. In this construction the bordism relates the homotopy quotient of a space by a circle action with its fixed point set via a correction involving the nontriviality of the induced circle bundle. Wilkins explained how this technology can apply to circle equivariant moduli space of holomorphic curves. This turns out to be quite involved since genericity prevents such spaces from being a homotopy Still the argument can be modified to produce a conceptual localization construction of Borman–Sheridan classes.
• Xu described his spectacular new joint work with Bai that proves the integral Arnold conjecture on the number of periodic orbits of a Hamiltonian vector field. Due to the existence of curves with nontrivial automorphisms, moduli spaces of stable maps behave like orbifolds rather than manifolds. Xu explained how the classical approach of Fukaya and Ono for extracting integral invariants from the moduli spaces can be made rigorous. The putative Fukaya–Ono invariants, which morally count curves with a trivial automorphism group, depend crucially on the stable complex structure on the moduli. Xu explained a new rigorous construction of such integral invariants in the case of genus zero symplectic Gromov–Witten theory and showed how the new invariants can be used to prove the hardest version of the Arnold conjecture.

Meeting Goals
The Homological Mirror Symmetry Annual Meeting will showcase progress made across the collaboration over the past five years. Speakers will present landmark results combining mirror constructions and proofs of categorical mirror statements as well as discuss unexpected and exciting applications of refined mirror constructions.
Talks will highlight progress on novel and deeper aspects of the homological mirror correspondence in two key areas. On the symplectic side, these include a SYZ constructions of mirrors of singularities and constructions of symplectic LandauGinzburg models, a categorical approach to virtual chains, a structure theory of Lagrangian surgeries, new implementations of the GrossSiebert intrinsic mirror symmetry program in the log CalabiYau setting, and a construction of integer valued GromovWitten invariants and new proofs of homological splitting
theorems. On the algebraic side, talks will explore the relation between cluster and representation theoretic mirror constructions and the GrossSiebert program, proofs of the Frobenius structure conjecture, and wall crossing and tropical vertex constructions of mirrors.The annual meeting will also investigate fundamental questions on the interaction of quantum spectra, quantum blowup formulas, and semiorthogonal decompositions of the derived categories of projective varieties. Finally, there will be discussions on nonarchimedean and tropical quantum invariants and their interactions with the SYZ and GrossSiebert mirror constructions.

Agenda
Thursday
9:30 AM Mohammed Abouzaid  A Decade of Mirror Symmetry 11:00 AM Abigail Ward  Symplectomorphisms Mirror to Birational Transformations of the Projective Plane 1:00 PM Tony Yue Yu  Mirror Structure Constants via NonArchimedean Analytic Disks 2:30 PM Jeffrey Hicks  Floer Theory and Lagrangian Cobordisms 4:00 PM Hang Yuan  Family Floer Mirror Space for Local SYZ Singularities Friday
9:30 AM Nicholas Wilkins  S^1 Localization via Pseudocycles, Applied to BormanSheridan Classes 11:00 AM Hulya Arguz  Fock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors 1:00 PM Guangbo Xu  Integral Count of Pseudoholomorphic Curves 
Abstracts
Mohammed Abouzaid
Columbia UniversityA Decade of Mirror Symmetry
Since the beginning of the HMS Simons collaboration, the areas of mirror symmetry has seen extensive developments, including the proof of longstanding conjectures, the introduction of new perspective on important problems and the opening of completely unexpected new connections. Mohammed Abouzaid will survey some if these developments, with a view towards where the potential future of the subject.
Hülya Argüz
University of GeorgiaFock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors
Cluster varieties come in pairs: for any Xcluster variety there is an associated Fock–Goncharov dual Acluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. Hulya Arguz will explain how to bridge the theory of cluster varieties with the algebrogeometric framework of Gross–Siebert mirror symmetry and show that the mirror to the Xcluster variety is a degeneration of the Fock–Goncharov dual Acluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Pierrick Bousseau.
Jeffrey Hicks
University of EdinburghFloer Theory and Lagrangian Cobordisms
A Lagrangian cobordism, as introduced by Arnold, is a Lagrangian submanifold with a notion of “ends” in the stabilization of a symplectic manifold X. The ends are Lagrangian submanifolds of X. The Floer theory of Lagrangian cobordisms was first studied by Biran and Cornea, who showed that the ends of a monotone Lagrangian cobordism are related by an iterated mapping cone in the Fukaya category.
Many known HMS correspondences rely on matching generators of categories. Provided we can meet the monotonicity requirement, Lagrangian cobordisms allow us to bootstrap to more general objects of the Fukaya category by exhibiting a Lagrangian cobordism from the generating Lagrangians to the desired object. In this talk, Jeffrey Hicks will discuss a few applications in HMS where this monotonicity requirement is met, what one can say beyond the monotone setting (unobstructedness) and how the geometry of Lagrangian cobordisms determines unobstructedness.
Abigail Ward
MITSymplectomorphisms Mirror to Birational Transformations of the Projective Plane
Abigail Ward will construct a nonfinite type fourdimensional Weinstein domain M_{univ} and describe an HMS correspondence between distinguished birational transformations of the projective plane preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense that it contains every Liouville manifold mirror to a log CalabiYau surface as a Weinstein subdomain; under Viterbo restriction to these subdomains, Ward will recover a mirror correspondence between the automorphism group of any open log CalabiYau surface and the symplectomorphism group of its mirror. We also give a conjectural HMS correspondence between the full group of volumepreserving birational transformations and a group of symplectomorphisms of nonexact symplectic deformations of M_{univ}. This is joint work in progress with Ailsa Keating.
Nicholas Wilkins
MITS^1 Localization via Pseudocycles, Applied to BormanSheridan Classes
S^1localisation due to AtiyahBott is a powerful tool to understand S^1equivariant cohomology, but historically it could not be leveraged against, for example, the S^1quotient of a moduli space parameterized by ES^1, as such a moduli space in general is not defined as a homotopy quotient for regularity reasons. In this talk, Wilkins will discuss a new way to interpret S^1localisation, constructed by way of a pseudocycle bordism between pseudocycles (assigned to an S^1manifold and its fixed point set) in equivariant homology. Wilkins will then demonstrate how to lift this method of S^1localisation to moduli spaces, and through this, prove a conjecture about BormanSheridan classes due to Seidel.
Guangbo Xu
Texas A&M UniversityIntegral Count of Pseudoholomorphic Curves
Due to the existence of curves with nontrivial automorphisms, moduli spaces of stable maps behave like orbifolds rather than manifolds. Therefore, invariants in symplectic geometry obtained from counting curves are generally rational numbers but not integers. In late 1990s, Fukaya and Ono sketched a new way to extract integral invariants from the moduli spaces. These invariants, which morally count curves with a trivial automorphism group, depend crucially on the stable complex structure on the moduli. Following FukayaOno proposal, in a recent preprint (2201.02688), we rigorously constructed such integral invariants in the case of genus zero symplectic GromovWitten theory. In this talk, Guangbo Xu will explain the original idea of FukayaOno, our technical construction, and potential extensions. This talk is based on the joint work with Shaoyun Bai.
Hang Yuan
Northwestern UniversityFamily Floer Mirror Space for Local SYZ Singularities
Hang Yuan will give an SYZ construction of singularities by the family Floer approach. The dual analytic fibration (including the singular fibers) will be explicitly presented in the talk. It modifies a nonArchimedean singular model of KontsevichSoibelman and also agrees with the work of GrossHackingKeelSiebert. Yuan will review the family Floer mirror construction and explain why it is necessary to challenge the conventional MaurerCartan picture for the sake of the nonArchimedean topology. By our new example, the dual singular fiber can be larger than the MaurerCartan set. The dual singular fiber also has an intrinsic geometric explanation by a classification of two types of quantum correction and the theory of ChambertLoir and Ducros. We will make a mathematically precise Tduality statement. We give evidence to convince people of it, even if people don’t know anything about the family Floer business. If time allowed, we will describe a version of SYZ converse and justify some computations for the folklore conjecture for the critical values of the mirror LandauGinzburg superpotential.
Tony Yue Yu
California Institute of TechnologyMirror Structure Constants via NonArchimedean Analytic Disks
For any smooth affine log CalabiYau variety U, Tony Yue Yu will construct the structure constants of the mirror algebra to U via counts of nonArchimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, Yu will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.