2022 Simons Collaboration on Homological Mirror Symmetry Annual Meeting

Date & Time


Location

Ingrid Daubechies Auditorium (IDA)

Thurs.: 8:30 AM—5 PM
Fri.: 8:30 AM—2 PM

Registration Closed

Invitation Only

Organizer:
Tony Pantev, University of Pennsylvania

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 Landau-Ginzburg models, a categorical approach to virtual chains, a structure theory of Lagrangian surgeries, new implementations of the Gross-Siebert intrinsic mirror symmetry program in the log Calabi-Yau setting, and a construction of integer valued Gromov-Witten 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 Gross-Siebert 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 blow-up formulas, and semi-orthogonal decompositions of the derived categories of projective varieties. Finally, there will be discussions on non-archimedean and tropical quantum invariants and their interactions with the SYZ and Gross-Siebert mirror constructions.

Previous meeting pages:

  • Agendaplus--large

    Thursday

    8:30 AMCHECK-IN & BREAKFAST
    9:30 AMMohammed Abouzaid | A Decade of Mirror Symmetry
    10:30 AMBREAK
    11:00 AMAbigail Ward | Symplectomorphisms Mirror to Birational Transformations of the Projective Plane
    12:00 PMLUNCH
    1:00 PMJeffrey Hicks | Floer Theory and Lagrangian Cobordisms
    2:00 PMBREAK
    2:30 PMTony Yue Yu | Mirror Structure Constants via Non-Archimedean Analytic Disks
    3:30 PMBREAK
    4:00 PMHang Yuan | Family Floer Mirror Space for Local SYZ Singularities
    5:00 PMDAY ONE CONCLUDES

    Friday

    8:30 AMCHECK-IN & BREAKFAST
    9:30 AMNicholas Wilkins | S^1 Localization via Pseudocycles, Applied to Borman-Sheridan Classes
    10:30 AMBREAK
    11:00 AMHulya Arguz | Fock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors
    12:00 PMLUNCH
    1:00 PMGuangbo Xu | Integral Count of Pseudoholomorphic Curves
    2:00 PMMEETING CONCLUDES
  • Abstractsplus--large

    Mohammed Abouzaid
    Columbia University

    A 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 long-standing 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 Georgia

    Fock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors

    Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster 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 algebro-geometric framework of Gross–Siebert mirror symmetry and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster 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 Edinburgh

    Floer 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
    MIT

    Symplectomorphisms Mirror to Birational Transformations of the Projective Plane

    Abigail Ward will construct a non-finite type four-dimensional 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 Calabi-Yau 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 Calabi-Yau surface and the symplectomorphism group of its mirror. We also give a conjectural HMS correspondence between the full group of volume-preserving birational transformations and a group of symplectomorphisms of non-exact symplectic deformations of M_{univ}. This is joint work in progress with Ailsa Keating.
     

    Nicholas Wilkins
    MIT

    S^1 Localization via Pseudocycles, Applied to Borman-Sheridan Classes

    In this talk, Nicholas Wilkins will discuss a new method (which is a generalization of certain related methods) to apply S^1-localisation using a pseudocycle bordism. By localization, we mean in the style of Atiyah-Bott, in that the bordism relates the S^1-homotopy quotient of a space with its fixed-point set, plus a term involving the non-triviality of the induced S^1-bundle. Applying this to an S^1-equivariant moduli space of holomorphic curves is not immediate, as genericity prevents such spaces from being a homotopy quotient, but Wilkins will demonstrate how to modify this argument in the example of Borman-Sheridan classes.
     

    Guangbo Xu
    Texas A&M University

    Integral 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 Fukaya-Ono proposal, in a recent preprint (2201.02688), we rigorously constructed such integral invariants in the case of genus zero symplectic Gromov-Witten theory. In this talk, Guangbo Xu will explain the original idea of Fukaya-Ono, our technical construction, and potential extensions. This talk is based on the joint work with Shaoyun Bai.
     

    Hang Yuan
    Northwestern University

    Family 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 non-Archimedean singular model of Kontsevich-Soibelman and also agrees with the work of Gross-Hacking-Keel-Siebert. Yuan will review the family Floer mirror construction and explain why it is necessary to challenge the conventional Maurer-Cartan picture for the sake of the non-Archimedean topology. By our new example, the dual singular fiber can be larger than the Maurer-Cartan set. The dual singular fiber also has an intrinsic geometric explanation by a classification of two types of quantum correction and the theory of Chambert-Loir and Ducros. We will make a mathematically precise T-duality 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 Landau-Ginzburg superpotential.
     

    Tony Yue Yu
    California Institute of Technology

    Mirror Structure Constants via Non-Archimedean Analytic Disks

    For any smooth affine log Calabi-Yau variety U, Tony Yue Yu will construct the structure constants of the mirror algebra to U via counts of non-Archimedean 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.

  • Participation & Fundingplus--large

    Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.

    Group A – PIs and Speakers
    The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Business-class or premium economy airfare will be booked for all flights over five hours.

    Group B – Out-of-town Participants
    The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.

    Group C – Local Participants
    Individuals in Group C are considered local and will not receive financial support, but are encouraged to enjoy all conference-hosted meals.

    Group D – Remote Participants
    Individuals in Group D will participate in the meeting remotely. Please register at the link above and a remote participation link will be sent to you approximately two weeks prior to the meeting.

  • Travel & Hotelplus--large

    Air and Train
    The foundation will arrange and pay for all air and train travel to the conference for those in Groups A and B. Please provide your travel specifications by clicking the registration link above. If you are unsure of your group, please refer to your invitation sent via email.

    Personal Car
    For participants in Groups A & B driving to Manhattan, The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.

    Hotel
    Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.

    The James NoMad Hotel
    22 E 29th St
    New York, NY 10016
    (between 28th and 29th Streets)
    https://www.jameshotels.com/new-york-nomad/

    For driving directions to The James NoMad, please click here.

  • COVID-19 Policyplus--large

    ALL in-person meeting attendees must be vaccinated against the COVID-19 virus with a World Health Organization approved vaccine, be beyond the 14-day inoculation period of their final dose, and provide proof of vaccination upon arrival to the conference. Acceptable vaccines can be found at the bottom of this page on the WHO’s site.

  • Reimbursementplus--large

    Individuals in Groups A & B will be reimbursed for meals not hosted by the Simons Foundation as well as local expenses, including ground transportation. Additional information in this regard will be emailed on the final day of the meeting.

  • Contactsplus--large

    Registration and Travel Assistance
    Claudio Proserpio
    Ovation Travel Group
    sfntravel@ovationtravel.com
    Phone: +1 646-998-9865
    Mobile: +1 646-320-2634
    www.ovationtravel.com

    Meeting Questions and Assistance
    Meghan Fazzi
    Manager, Events and Administration, MPS, Simons Foundation
    mfazzi@simonsfoundation.org
    (212) 524-6080

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