Simons Collaboration on Homological Mirror Symmetry

Schedule

Friday December 4

Saturday December 5

Titles and abstracts

Denis Auroux: Lagrangian Floer theory for trivalent graphs and HMS for curves

Abstract: The mirror of a genus g curve can be viewed as a trivalent configuration of 3g−3 rational curves meeting in 2g−2 triple points; more precisely, this singular configuration arises as the critical locus of the superpotential in a 3-dimensional Landau-Ginzburg mirror. In joint work with Alexander Efimov and Ludmil Katzarkov, we introduce a notion of Fukaya category for such a configuration of rational curves, where objects are embedded graphs with trivalent vertices at the triple points, and morphisms are linear combinations of intersection points as in usual Floer theory. We will describe the proposed construction of the structure maps of these Fukaya categories, attempt to provide some motivation, and outline examples of calculations that can be carried out to verify homological mirror symmetry in this setting.

Mandy Cheung: Polytopes, wall crossings, and cluster varieties

Abstract: Cluster varieties are log Calabi-Yau varieties which are a union of algebraic tori glued by birational “mutation” maps. Partial compactifications of the varieties, studied by Gross, Hacking, Keel, and Kontsevich, generalize the polytope construction of toric varieties. However, it is not clear from the definitions how to characterize the polytopes giving compactifications of cluster varieties. We will show how to describe the compactifications easily by broken line convexity. As an application, we will see the non-integral vertex in the Newton Okounkov body of Gr(3,6) comes from broken line convexity. Further, we will also see certain positive polytopes will give us hints about the Batyrev mirror in the cluster setting. The mutations of the polytopes will be related to the almost toric fibration from the symplectic point of view. Finally, we can see how to extend the idea of gluing of tori in Floer theory which then ended up with the Family Floer Mirror for the del Pezzo surfaces of degree 5 and 6. The talk will be based on a series of joint works with Bossinger, Lin, Magee, Najera-Chavez, and Vianna.

Ron Donagi: On Integrable Systems and 3D Mirror Symmetry

Abstract: We explore some aspects of 3D Mirror Symmetry between Coulomb and Higgs branches for super conformal field theories of class S, and the related tamely ramified meromorphic Hitchin systems. A key geometric ingredient is the construction of a family of Hitchin systems over the Deligne-Mumford compactification. We will also explore connections to the Deligne-Simpson problem and the behavior of good, bad and ugly theories.

Kenji Fukaya: Symmetry and genericity: Works in progress on VFC

Abstract: In this talk I want to report certain aspects of the application of virtual fundamental chain technique to pseudo holomorphic curves. The basic package have been established. However to apply it in various concrete situations several issues appear. The typical problem is how to work out the perturbation while keeping the symmetry of the problem. I want to explain several situations such issue occurs and various method to resolve it.

Sheel Ganatra: Rabinowitz wrapped Fukaya categories and the categorical formal punctured neighborhood of infinity

Abstract: The Rabinowitz (wrapped) Fukaya category is a categorical invariant of exact Lagrangians in a Liouville manifold whose morphisms (“Rabinowitz wrapped Floer homology groups”) measure the failure of Poincare duality in wrapped Floer theory. We will describe some general properties of this category and, following a conjecture of Abouzaid, relate the Rabinowitz and usual wrapped Fukaya categories by way of a general construction introduced by Efimov, the “categorical formal punctured neighborhood of infinity”. Using this result, we show how Rabinowitz Fukaya categories can be fit into – and therefore computed in terms of – mirror symmetry. Joint work (in progress) with Yuan Gao and Sara Venkatesh.

Mark Gross: The canonical scattering diagram

Abstract: Given a log Calabi-Yau with maximally degenerate boundary or given a maximally unipotent degeneration of Calabi-Yau manifolds, one may construct the mirror family using the data of a “scattering diagram”. This was carried out by Kontsevich and Soibelman in dimension 2 for K3 surfaces in 2004, generalized to toric degenerations in all dimensions by myself and Siebert in 2007. Both of these approaches used an algorithmic method of constructing the scattering diagram. However, work of myself, Hacking and Keel (2011) introduced the “canonical scattering diagram” for constructing the mirror to a log Calabi-Yau surface, which was defined using relative Gromov-Witten theory of the surface. In this talk, I discuss joint work with Siebert generalising this construction to all dimensions, using the punctured invariants of Abramovich-Chen-Gross-Siebert.

Maxim Kontsevich: Towards dimension theory for spectral semi-orthogonal decompositions

Abstract: It is expected that the derived category of coherent sheaves on a smooth projective variety admits a family of semiorthogonal decompositions governed by the generic spectrum of quantum multiplication deformed by algebraic classes. I present a hypothetical formula for the Serre dimension of elementary pieces, and sketch applications to the rationality questions.

Yin Li: Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres

Abstract: It is an interesting question whether an odd-dimensional spherical object can be a phantom in a triangulated category. In symplectic topology, a related question is whether any odd-dimensional Lagrangian sphere in a Weinstein manifold has non-trivial homology class? We show that this is the case for many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi-Yau in the sense of Keller, and provide examples of such manifolds using Koszul duality between A_infty algebras.

Cheuk Yu Mak: Fukaya-Seidel category, Hilbert scheme and category O

Abstract: Khovanov homology is a link invariant which categorifies the Jones polynomial. Various definitions of Khovanov homology have been given by seemingly unrelated methods, including representation theory, symplectic
geometry and algebraic geometry. In this talk, we will review how the symplectic version of Khovanov homology is defined. Subsequently, we will describe a distinguished Lefschetz fibration on the symplectic manifold and its associated Fukaya-Seidel category. Finally, we will give some applications on annular Khovanov homology. This is a joint work with Ivan Smith.

Abigail Ward: An approach to homological mirror symmetry for log Calabi-Yau surfaces

Abstract: We will discuss an approach towards proving homological symmetry for log Calabi-Yau surfaces (Y,D) which gives an explicit correspondence between line bundles on Y and monomially admissible Lagrangian sections of an SYZ fibration over an integral affine base with focus-focus singularities. The base of this fibration is a deformation of the integral affine base that appears in the mirror construction of Gross, Hacking, and Keel; the intersection points of the Lagrangian sections can be identified with theta functions which appear in the Gross-Hacking-Keel construction and in subsequent work. We will also discuss how this approach may generalize to higher dimensional log Calabi-Yau pairs with toric models. This is joint work in progress with Andrew Hanlon.