The CMSA will be hosting a four-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on January 10-13, 2018. The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.
Please click here to register for this event. We have space for up to 30 registrants on a first come, first serve basis.
Confirmed Participants:
Wednesday, January 10
| Time | Speaker | Title/Abstract |
| 9:30-10:30am | Tony Pantev | Homological Mirror Symmetry and the mirror map for del Pezzo surfaces
Abstract: I will discuss the general mirror symmetry question for |
| 10:30 – 11:00am | Break | |
| 11:00 – 12:00pm | Young-Hoon Kiem | Knorrer periodicity in curve counting
Abstract: The derived Knorrer periodicity compares the derived category of coherent sheaves on a projective hypersurface with that of matrix factorizations of its defining equation. I’d like to talk about a parallel development in curve counting, including Chang-Li’s p-field invariant, Chang-Li-Li’s algebraic theory of (narrow) FJRW invariant and Polishchuk-Vaintrob’s cohomological field theory, from the viewpoint of cosection localization. |
| 12:00 – 1:45pm | Lunch | |
| 1:45 – 2:45pm | Kaoru Ono | Anti-symplectic involutions and twisted sectors in Langranian Floer theory
Abstract: After explaining some results in Lagrangian Floer theory in the presence of an anti-symplectic involution, I will present a definition of twisted sectors, which is suitable for Lagrangian Floer theory in orbifold setting. The first part is based on joint works with K. Fukaya, Y.-G. Oh and H. Ohta and the second is based on a joint work (in progress) with B. Chen and B.-L. Wang. |
| 2:45 – 3:15pm | Tea | |
| 3:15 – 4:15pm | Radu Laza | Some remarks on degenerations of K-trivial varieties
Abstract A fundamental result for K3 surfaces is the Kulikov-Persson-Pinkham theorem on degenerations of K3 surfaces. In this talk, I will explore higher di mensional analogues of it and potential applications. Specifically, as a consequence of the minimal model program, Fujino has a obtained a weak analogue of the KPP Theorem for K-trivial varieties. I will then discuss some relationships between the dual complex of the central fiber and the monodromy of the degenerations. I will then explain some consequences of this for Hyperkaehler manifolds and Calabi-Yau 3-folds. |
Thursday, January 11
| Time | Speaker | Title/Abstract |
| 9:30-10:30am | Yan Soibelman |
Riemann-Hilbert correspondence in dimension one, Fukaya categories and periodic monopoles Abstract: By RH-correspondence in dimension one I understand not only the classical one for holonomic D-modules on curves, but also its versions for q-difference and elliptic difference equations. The unifying geometry for all versions is the one of partially compactified symplectic surfaces. Then the RH-correspondence relates the category of holonomic coherent sheaves on the quantized symplectic surface with an appropriate partially wrapped Fukaya category of that surface. The non-abelian Hogde theory in dimension one deals with twistor families of the parabolic versions of the above categories. In the case of q-difference equations the role of harmonic objects is played by doubly periodic monopoles, while in the case of elliptic difference equations it is played by triply periodic monopoles. Talk is based on the joint project with Maxim Kontsevich.
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| 10:30 – 11:00am | Break | |
| 11:00 – 12:00pm | Cheol-Hyun Cho | Gluing localized mirror functors.
Abstract: Given a Lagrangian submanifold L, we can consider a formal deformation theory of $L$ which is developed by Fukaya-Oh-Ohta-Ono. This provides a local mirror (with respect to L), given by the Lagrangian Floer potential function on the formal Maurer-Cartan space of L. Then, we can canonically construct a localized mirror functor from Fukaya category to the matrix factorization category. Given two different Lagrangian submanifolds, we explain how to glue these local mirrors to obtain a global mirror model, and also how to glue their localized mirror functors to obtain a global version of homological mirror functor. This is a joint work in progress with Hansol Hong and Siu-Cheong Lau. |
| 12:00 – 1:45pm | Lunch | |
| 1:45 – 2:45pm | Mohammed Abouzaid | |
| 2:45 – 3:15pm | Tea | |
| 3:15 – 4:15pm | Siu-Cheong Lau | Immersed Lagrangians and wall-crossing
Abstract: We find the Floer-theoretical gluing between local moduli of Lagrangian immersions, and use it to study wall-crossing for local Calabi-Yau manifolds. It is a joint work with Cho and Hong. In a joint work with Hong and Kim, we apply the technique to recover the Lie theoretical mirror of Gr(2,n). |
Friday, January 12
| Time | Speaker | Title/Abstract |
| 9:30-10:30am | Eric Zaslow | Framing Duality
Abstract: A symmetric quiver with g nodes is described by a symmetric adjacency matrix of size g. The same data defines a “framing” of a certain genus-g Legendrian surface in the five-sphere, and the invariants of the quiver conjecturally relate to the open Gromov-Witten (GW) invariants of a non-exact Lagrangian filling of the surface. (Physically, both data count the same BPS states but from different perspectives.) Further, cluster theory can be exploited to conjecturally obtain all open GW invariants of Lagrangian fillings of a wider class of Legendrian surfaces described by cubic planar graphs.
In this talk, I will describe these observations, which build on prior work of others and are explored in joint works with David Treumann and Linhui Shen. |
| 10:30 – 11:00am | Break | |
| 11:00 – 12:00pm | Si Li | Calabi-Yau geometry, Kodaira-Spencer gravity and integrable hierarchy
Abstract: We discuss some physical and geometric aspects of Kodaira-Spencer gravity (BCOV theory) on Calabi-Yau geometry and explain how quantum master equation leads to integrable hierarchies |
| 12:00 – 1:45pm | Lunch | |
| 1:45 – 2:45pm | Sergueï Barannikov | Quantum master equation on cyclic cochains and categorical higher genus Gromov-Witten invariants
The construction of cohomology classes in the compactified moduli spaces of curves based on the quantum master equation on cyclic cochains will be reviewed. For the simplest category consisting of one object with only the identity morphism it produces the generating function for products of the psi-classes. The talk is based on the speaker’s works “Modular operads and Batalin-Vilkovisky geometry” (MPIM Bonn preprint 2006-48 (04/2006)) and “Noncommutative Batalin–Vilkovisky geometry and matrix integrals” (preprint Hal-00102085 (09/2006)). |
| 2:45 – 3:15pm | Tea | |
| 3:15 – 4:15pm | Thomas Lam | Mirror symmetry for flag varieties via the Langlands program
Abstract: I will talk about a mirror theorem for minuscule flag |
| 4:15 – 4:30pm | Break | |
| 4:30 – 5:30pm | Colleen Robles
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Generalizing the Satake-Baily-Borel compactification.
Abstract: The Satake-Baily-Borel (SBB) compactification is an projective algebraic completion of a locally Hermitian symmetric space. This construction, along with Borel’s Extension Theorem, provides the conduit to apply Hodge theory to study the moduli spaces (and their compactifications) of principally polarized abelian varieties and K3 surfaces. Most period domains are not Hermitian, and so one would like to generalize SBB in the hopes of similarly applying Hodge theory to study the moduli spaces (and their compactifications) of more general classes of algebraic varieties. In this talk I will present one such generalization. This work joint work with M. Green, P. Griffiths and R. Laza. |
Saturday, January 13
| Time | Speaker | Title/Abstract |
| 9:30-10:30am | Chenglong Yu | Higher Hasse-Witt matrices and period integrals
Abstract: I shall explain a program to relate the arithmetic of Calabi-Yau hypersurfaces in toric varieties or flag varieties, to their period integrals at the large complex structure limit. In particular, we prove a recent conjecture of Vlasenko regarding higher Hasse-Witt matrices. This work follows Katz’s description of Frobenius action in terms of local expansions. It is joint work with Huang, Lian and Yau.
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| 10:30 – 11:00am | Break | |
| 11:00 – 12:00pm | Kazushi Ueda | Moduli of K3 surfaces as moduli of A-infinity structures
Abstract: We give a description of the moduli space of K3 surfaces polarized |
