# Simons Collaboration Workshop, April 5-7, 2018

The CMSA will be hosting a three-day Simons Collaboration Workshop on Homological Mirror Symmetry and Hodge Theory on April 5-7, 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.

We may be able to provide some financial support for grad students and postdocs interested in this event.  If you are interested in funding, please send a letter of support from your mentor to Hansol Hong at hansol84@gmail.com.

Confirmed Speakers:

The schedule is as follows:

Thursday 4/5/2018

 Time Speaker Title/Abstract 12:00-1:30pm Lunch 1:30-2:30pm Tristan Collins Title: BPS B-branes and stability Abstract:  I will give a short introduction to the deformed Hermitian-Yang-Mills equation and  discuss the (conjectural/motivational) relationship with stability in the sense of Bridgeland. This talk will cover joint work with A. Jacob, D. Xie, and S.-T. Yau. 2:30-2:45pm Break 2:45-3:45pm Dimitry Vaintrob Title: Operads and circle actions Abstract: Cohomology of the topological operad FLD of framed little disks (a.k.a. the BV operad) acts on the Hochschild homology of any Calabi-Yau algebra. Cohomology of the related topological operad of marked nodal genus zero curves acts on a deformation of the cohomology of any symplectic manifold, and this action is responsible for all quantum product operations. It was proven by Bruno Vallette and Drummond Cole that an action of $\mathbb{Q}$-homology of the operad of marked nodal curves is equivalent, in genus zero, to an action of the homology of the operad of framed little disks together with a trivialization, in a homotopy-theoretic sense, of the BV operator $\Delta$. In a later paper Drummond-Cole showed that this result holds in a certain category of topological operads, so that in particular it is also true (on the dg level) for cohomology with coefficients in $\mathbb{Z}$ (or in an arbitrary field). In work with Alex Oancea we give a higher genus version of this result, using Segal moduli spaces of curves with parametrized boundary and their compactifications. Time permitting, I will also mention certain motivic enhancements of our result (based on more recent work) which give compatibility with Galois actions and de Rham lattices on the two sides, a result already new in genus zero. 3:45-4:15pm Break 4:15-5:15pm Mandy Cheung Title: Counting tropical curves by quiver representation Abstract: The work of Gross-Hacking-Keel-Kontsevich tell the relation between scattering diagrams and cluster algebras. In the talk, we will describe those objects with quiver representations. After that, we will give a expression of tropical curves counting by quiver representations. This is a joint work in progress with Travis Mandel.

Friday 4/6/2018

 Time Speaker Title/Abstract 9:00 – 9:30am Breakfast 9:30-10:30 am Zack Sylvan 10:30-11:00am Break 11:00-12:00pm Yu Pan Title: Augmentations categories and exact Lagrangian cobordisms. Abstract: To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between  two Legendrianknots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomologyof morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of  objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials. 12:00-1:30pm Lunch 1:30-2:30pm Cheuk-Yu Mak Title: Tropically constructed Lagrangians in mirror quintic threefolds Abstract: In this talk, we will explain how to construct closed Lagrangian submanifolds in mirror quintic threefolds using tropical curves and the toricdegeneration technique. As an example, we will illustrate how the corresponding Lagrangians look like for tropical curves that contribute to the Gromov–Witteninvariant of the line class of the quintic threefold. We will also show that multiplicity of a tropical curve, in this symplectic setting, will be realized as the order of the torsion the first homology group of the Lagrangian. This is a joint work with Helge Ruddat. 2:30-2:45pm Break 2:45-3:45pm Yu-Shen Lin 3:45-4:15pm Break 4:15-5:15pm Yoosik Kim Title: Mirror construction of Grassmannians via immersed Lagrangian Floer theory Abstract: A partial flag manifold admits a completely integrable system, so-called a Gelfand-Cetlin system, constructed by Guillemin-Sternberg. The fibers of the system are almost like toric fibers. However, as the big torus action does not extend to boundary strata, non-toric Lagrangian fibers may appear at a boundary stratum. In the first part of the talk, we classify all Lagrangian fibers on partial flag manifolds of various types. After discussing it, we exhibit a construction of a mirror of some low dimensional Grassmannians using Strominger-Yau-Zaslow mirror symmetry. To incorporate non-toric Lagrangian fibers, which are sometimes non-zero objects in the Fukaya category, we produce immersed Lagrangians arising from smoothing faces containing a face having non-toric Lagrangian. We then glue deformation spaces of Lagrangians to obtain the Rietsch mirror. This talk is based on joint work with Yunhyung Cho and Yong-Geun Oh, and ongoing joint work with Hansol Hong and Siu-Cheong Lau.

Saturday 4/7/2018

 Time Speaker Title/Abstract 8:30-9:00am Breakfast 9:00-10:00am Jacob Bourjaily Title: Stratifying On-Shell Cluster Varieties   Abstract: There exists a deep correspondence between a class of physically important functions—called “on-shell functions”—and certain (cluster variety) subspaces of Grassmannian manifolds, endowed with a volume form that is left invariant under cluster coordinate transformations. These are called “on-shell varieties” (which may or may not include all cluster varieties). It is easy to prove that the number of on-shell varieties is finite, from which it follows that the same is true for on-shell functions. This is powerful and surprising for physics, because these on-shell functions encode complete information about perturbative quantum field theory. In this talk, I describe the details of this correspondence and how it is constructed and give the broad physics motivations for obtaining a more systematic understanding of on-shell cluster varieties. I outline a general, brute-force strategy for classifying these spaces; and describe the results found by applying this strategy to the case of Gr(3,6). 10:00-10:15am Break 10:15-11:15am Shu-Heng Shao Title: Vertex Operator Algebra, Wall-Crossing Invariants, and Physics Abstract: Motivated by four-dimensional conformal field theory with N=2 supersymmetry, we discuss an interesting relation between vertex operator algebras (VOAs) and Kontsevich-Soibelman wall-crossing. We discuss a conjectured formula for the vacuum character of this VOA from the associated Kontsevich-Soibelman wall-crossing invariant of the four-dimensional field theory. We further generalize this proposal to include extended supersymmetric objects, known as line defects and surface defects, into the four-dimensional field theory. Each such defect gives rise to a module of the associated VOA and we propose a formula for the character of this module. 11:15-11:30am Break 11:30-12:30pm Mauricio Romo Title: Aspects of B-twisted (2,2) and (0,2) hybrid models Abstract: I will talk about properties and definition of certain sphere correlators for elements on the chiral ring of B-twisted hybrid models for the case they posses (2,2) and (0,2) supersymmetry. I will review these models and their B-chiral ring. I will present some interesting analytic properties of these correlators and some sufficient criteria for the absence of instanton corrections in the (0,2) case.