- Organized by
Tony Pantev, Ph.D.University of Pennsylvania
The fourth annual meeting of the Simons Collaboration on Homological Mirror Symmetry will focus on recent work extracting deep geometric and arithmetic statements from the homological mirror interactions between symplectic topology, Hodge theory, non-commutative geometry and differential geometry. Talks will highlight progress on the motivic and cycle theoretic consequences of recently developed arithmetic enhancements of Floer theory and will explore constructive and birational geometry aspects of tropical and quantum invariants over non-algebraically closed fields. Additionally talks focused on differential geometry will showcase novel existence results for special Lagrangian tori and their implications for the homological mirror correspondence.
Collaboration website: https://schms.math.berkeley.edu/
Thursday, November 14
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Nick Sheridan | Lagrangian Cobordism and Chow groups 10:30 AM BREAK 11:00 AM Umut Varoulgunes | Local-to-Global Principles in Floer Theory 12:00 PM LUNCH 1:00 PM Denis Auroux | Coisotropic Branes and HMS for Tori 2:00 PM BREAK 2:30 PM Konstanze Rietsch | The Tropical Critical Point and Mirror Symmetry 3:30 PM BREAK 4:00 PM Tristan Collins | Special Lagrangians in Tian-Yau Manifolds and SYZ Mirror Symmetry 5:00 PM DAY ONE CONCLUDES
Friday, November 15
8:30 AM CHECK-IN & BREAKFAST 9:30 AM Paul Seidel | Automorphisms of Fukaya Categories and Quantum Steenrod Operations 10:30 AM BREAK 11:00 AM Mark Mclean | Birational Calabi-Yau Manifolds have the Same Small Quantum Products 12:00 PM LUNCH 1:00 PM Shing-Tung Yau | The HMS/SYZ Program: A Progress Report 2:00 PM MEETING CONCLUDES
University of California, Berkeley
Coisotropic Branes and HMS for Tori
Auroux will describe an approach, developed in the Ph.D. thesis of his student Yingdi Qin, to the problem of incorporating coisotropic branes into the Fukaya category of a torus, as well as the motivation for this construction from the perspective of SYZ and homological mirror symmetry. Qin’s work also gives insight into the equivalence between the Fukaya categories of dual symplectic tori (e.g., elliptic curves with inverse areas), which Auroux will explain if time permits.
Massachusetts Institute of Technology
Special Lagrangians in Tian-Yau Manifolds and SYZ Mirror Symmetry
Collins will discuss the existence of special Lagrangian submanifolds in noncompact Calabi-Yau manifolds constructed by Tian-Yau and connections with mirror symmetry. Collins will discuss the existence of special Lagrangian torus fibrations in complex dimension 2 on Tian-Yau surfaces and a conjectural dual special Lagrangian fibration on the mirror surface. This is joint work with A. Jacob and Y.-S. Lin.
Stony Brook University
Birational Calabi-Yau Manifolds Have the Same Small Quantum Products
We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of symplectic cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.
King’s College London
The Tropical Critical Point and Mirror Symmetry
Call a (generalized) Puiseaux series positive if the leading term is a positive real number. Suppose we are given a Laurent polynomial f(x_1,…, x_n) over the field of generalized Puiseaux series, and that f has positive coefficients. Rietsch shows that, under a mild hypothesis on the Newton polytope, such a Laurent polynomial has a unique positive critical point. Rietsch will give some applications of this result related to mirror symmetry. This is joint work with Jamie Judd.
Massachusetts Institute of Technology
Automorphisms of Fukaya Categories and Quantum Steenrod Operations
Seidel will consider the formal germ of the autoequivalence group of a Fukaya category. This contains arithmetic information, partly encoded in quantum Steenrod operations.
University of Edinburgh
Lagrangian Cobordism and Chow Groups
Homological mirror symmetry implies an isomorphism between the Grothendieck group of the derived category and that of the Fukaya category. The former is related to the Chow group, via the Chern character, whereas the latter is related to the Lagrangian cobordism group through the work of Biran-Cornea. One can try to compare these two groups directly. Sheridan will describe joint work with Ivan Smith in which they take some preliminary steps in this direction and find some interesting analogies.
Local-to-Global Principles in Floer Theory
Varoulgunes will start by reviewing his thesis work, which introduced Floer theoretic invariants for arbitrary compact subsets of symplectic manifolds and proved a descent statement under restrictive assumptions. He will then report on ongoing work regarding applications to symplectic manifolds that are presented as the total space of a Lagrangian fibration or the union of a standard neighborhood of a symplectic SC divisor and a Liouville domain. These are joint works with D. Tonkonog, M. Abouzaid and Y. Groman.
The HMS/SYZ Program: A Progress Report
Yau will survey recent progress, with particular focus on the dHYM approach to stability, mirror functors by gluing construction and construction of a new class of singular CY mirror pairs by cyclic covers, as well as their applications.
Air and Train
Groups A & BThe 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.
Group CIndividuals in Group C will not receive financial support. Please register at the link above so we can capture your dietary requirements. If you are unsure of your group, please refer to your invitation sent via email.
Personal CarFor participants in Groups A & B driving to Manhattan, the Roger 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.
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The Roger hotel. Any additional nights are at the attendee’s own expense.
The Roger New York
131 Madison Avenue
New York, NY 10016
(between 30th and 31st Streets)
To arrange accommodations, please register at the link above.
For driving directions to The Roger, please click here.