Workshop on Homological Mirror Symmetry: methods and structures

November 7-11, 2016

** All talks will take place in Wolfensohn Hall

Monday, 11/7

10:00 am - 11:00 am Hiroshi Iritani, Kyoto University, "On the Gamma conjecture associated with toric flips"

Abstract: Gamma conjecture in a broad sense predicts a relationship between decompositions of quantum differential equations and those of derived categories. In this talk, I will discuss a partially compactified Landau-Ginzburg mirror symmetry for toric stacks. This picture naturally leads to a formal decomposition of the quantum differential equations of toric stacks under flips. Part of this talk is based on joint work with Tom Coates, Alessio Corti and Hsian-Hua Tseng.

11:30 am - 12:30 pm Ailsa Keating, IAS, "On symplectomorphism groups of some Milnor fibres"
12:30 pm - 2:30 pm LUNCH, Dining Hall
2:30 pm - 3:30 pm Alex Perry, Harvard, "Categorical joins"

Abstract: Homological projective duality is a powerful theory developed by Kuznetsov for studying the derived categories of varieties. It can be thought of as a categorification of classical projective duality. I will describe a categorification of the classical join of two projective varieties, its relation to homological projective duality, and applications to the derived categories of some Fano and Calabi-Yau varieties. This is joint work with Alexander Kuznetsov.

3:30 pm - 4:00 pm TEA, Front of Wolfensohn Hall
4:00 pm - 5:00 pm Lenhard Ng, Duke, "Knot contact homology and partially wrapped Floer homology"

Abstract: I'll describe how to prove that the conormal torus is a complete knot invariant, via holomorphic curves and an enhanced form of knot contact homology. The proof is motivated by an isomorphism between a partially wrapped Floer homology defined in this context and the group ring of the knot group. This is joint work with Tobias Ekholm and Vivek Shende.


Tuesday, 11/8

10:00 am - 11:00 am Daniel Huybrechts, University of Bonn, "Kuznetsov's Calabi-Yau categories: introduction and applications"

Abstract: The talk will start with a gentle introduction into Kuznetsov's construction of (fractional) Calabi-Yau categories associated with hypersurfaces. The relation to matrix factorizations will be mentioned, but the talk will mainly focus on the Fourier-Mukai aspects. As an application we will discuss a new proof of the Global Torelli theorem for cubic fourfolds (joint work with Jorgen Rennemo).

11:30 am - 12:30 pm John Calabrese, Rice University, "TBA"
12:30 pm - 2:30 pm LUNCH, Dining Hall
3:30 pm - 4:00 pm TEA, Front of Wolfensohn Hall

Wednesday, 11/9

10:00 am - 11:00 am Emmy Murphy, MIT, "Mirror symmetry for the trefoil knot"

Abstract: We will compute the wrapped Fukaya category of the affine variety {xyz+x+z=0}, in particular showing that it is self mirror. The result isn't new, but the methods are. The tools used are Legendrian contact homology and a bunch of soft techniques. Hopefully the talk will illustrate why the tools work for general affine varieties of arbitrary dimension, without induction on dimension.

11:30 am - 12:30 pm Jingyu Zhao, IAS, "Periodic symplectic cohomology and the Hodge filtration"

Abstract: For an open symplectic manifold, the homological mirror symmetry conjecture states that there is an derived equivalence between the wrapped Fukaya category of the symplectic manifold and the category of matrix factorizations of its mirror Landau-Ginzburg model. It is conjectured in the work of Kontsevich-Katzarkov-Pantev that the periodic cyclic homology of a smooth and proper DG-category admits a non-commutative Hodge structure. Motivated by this, in this talk we define the periodic cyclic homology of the wrapped Fukaya category. Due to the non-properness of the wrapped Fukaya category, the usual definition of periodic cyclic homology is not well-behaved with respect to localization. To resolve this, we propose another definition, called periodic symplectic cohomology, and define the corresponding Hodge filtration on it.

12:30 pm - 2:30 pm LUNCH, Dining Hall
2:30 pm - 3:30 pm Nick Sheridan, IAS, "Versality for the relative Fukaya category"
3:30 pm - 4:00 pm TEA, Front of Wolfensohn Hall
4:00 pm - 5:00 pm Ivan Smith, University of Cambridge, "Mirror symmetry for the mirror quartic, and other stories"

Thursday, 11/10

10:00 am - 11:00 am Fabian Haiden, Harvard, "Flow on quiver representations, nested logarithms, and weight filtrations in artinian categories"
11:30 am - 12:30 pm Zach Sylvan, IAS, "Partially wrapped Floer theory"

Abstract: I'll define a version of Floer theory associated to a Liouville domain with "stops", which are Liouville hypersurfaces of the boundary. I'll then use this to describe the Viterbo restriction maps on wrapped Fukaya categories and explain what the new description has to say about mirrors to reducible divisors.

12:30 pm - 2:30 pm LUNCH, Dining Hall
2:30 pm - 3:30 pm Sheel Ganatra, IAS, "Localizing the Fukaya category of a Weinstein manifold"
3:30 pm - 4:00 pm TEA, Front of Wolfensohn Hall
4:00 pm - 5:00 pm Dmitry Tamarkin, Northwestern University, "Microlocal category over divided powers"

Abstract: In my preprint the microlocal category is defined over the Novikov ring with rational coefficients. I will explain how to modify the construction so as to define the microlocal category over the divided power version of Novikov ring over integers, where the coefficient of T^c is of denominator factorial of the floor of c.


Friday, 11/11

10:00 am - 11:00 am Kenji Fukaya, Stonybrook University, "Open closed and closed open map revisited"
11:30 am - 12:30 pm Mohammed Abouzaid, IAS, "Family Floer theory and mirror symmetry"
12:30 pm - 2:30 pm LUNCH, Dining Hall

End of workshop