# Who is Who in Mirror Symmetry

May 27–June 1, 2019

### Mohammed Abouzaid

###### Curved A-infinity ring spectra

In order to be able to formulate Floer homotopy type of Lagrangian submanifolds, one needs a notion of curved algebras in the setting of stable homotopy theory. I will formulate one such notion. This is joint work in progress with A. Blumberg and T. Kragh.

### Serguei Barannikov

###### First part: Invariants of Morse complexes and persistent homology.(20min) Second part: Higher genus GW: associative algebra Q(N) and the total generating function for psi-classes and their products.(40min)

The talk consists of two parts.

1st part: There is canonical partition of set of critical values of smooth function into pairs "birth-death" and a separate set representing basis in Betti homology, as was established in the speaker's paper in 1994. This partition arises from bringing the Morse complex to so called "canonical form" by a linear transform respecting the filtration defined by the order of the critical values. These "canonical forms" are combinatorial invariants of filtered complexes. Starting from the beginning of 2000s these invariants became widely used in applied mathematics under the name of "Persistence diagrams" and "Persistence Bar-codes". Currently there are over 5000 articles of more than 500 researchers applying these invariants in different disciplines ranging from symplectic geometry, biology, cosmology to artificial neural nets.

2nd part: The speaker's approach to the categorical construction of higher genus Gromov-Witten invariants is reviewed in the simplest case. The ncBV formalism over the associative algebra Q(N) gives explicit combinatorial formula for the total cohomology $$H^*(\bar{M}_{g,n})$$- valued generating function for $$\psi$$ classes and their products.

References: arXiv:1803.11549 (HAL-00429963 (2009)), arXiv:1710.08442 (Preprint MPIM(Bonn) 2006-48 (04/2006))

### Roman Bezrukavnikov

###### Mirror symmetry and local geometric Langlands

I will describe a joint project with Michael McBreen aimed at establishing homological mirror symmetry where the B-side is a local Calabi-Yau (moreover, homolorphically symplectic) variety such as the cotangent bundle of the flag space. Time permitting I will discuss relation to Bridgeland stabilities and to representation theory of quantum groups at a root of unity and affine Lie algebras at the critical level.

### Agnieszka Bodzenta

###### Exact categories and abelian envelopes

For exact categories I will develop a theory parallel to the theory well-known for triangulated categories; left and right admissible subcategories, and (semi-orthogonal) decompositions. In particular, I will introduce thin exact categories, i.e. exact categories with full exceptional collections. I will discuss left and right abelian envelopes of an exact category and will show that highest weight categories are precisely the abelian envelopes of thin exact categories. I will also discuss Ringel duality from this point of view. This is joint work with A. Bondal.

### Alexey Bondal

###### Categorical approach to ranks of tensors of valency 3

We will discuss a way to define ranks of objects in exact, abelian or triangulated categories which stems from the complexity theory.

### Tristan Collins

###### Nonlinear PDE and stabilities for Categoreis

I will discuss an infinite dimensional GIT approach to stable holomorphic bundles and connections with Bridgeland type stability conditions. This leads to algebraic obstructions to existence of special Lagrangians in Landau–Ginzburg models mirror to toric Fano manifolds.

###### Some results in SYZ mirror symmetry

I will discuss two recent results motivated by SYZ mirror symmetry. First, I will discuss the existence of special Lagrangian submanifolds in complete, non-compact Calabi-Yau manifolds, and the existence of SYZ fibrations of complete CY surfaces. Secondly, I will discuss an infinite dimensional GIT approach to stable holomorphic bundles and connections with Bridgeland type stability conditions. This leads to algebraic obstructions to existence of special Lagrangians in Landau-Ginzburg models mirror to toric Fano manifolds. This talk will discuss joint works with A. Jacob, Y.-S. Lin, and S.-T. Yau.

### Alessio Corti

###### Mirror symmetry and smoothing toric Fano 3-folds

I state a one-to-one correspondence between the set of families of Fano 3-folds and the set of mutation classes of certain Laurent polynomials. I sketch a proof of this statement. I also state a conjecture on deformations of Gorenstein affine toric 3-folds.

### Alexander Efimov

###### Cheeger–Simons invariants of canonical extensions are torsion

We will sketch a generalization of the results of Iyer and Simpson arXiv:0707.0372 to the general case of a divisor with simple normal crossings.

### Kenji Fukaya

###### Floer homology of noncompact Lagrangian submanifold in the divisor complement

In my previous paper with A. Daemi we discussed Floer homology of compact Lagrangian submanifold of $$X$$ minus $$D$$ where $$D$$ is a smooth divisor of $$X$$. In this talk I would like to explain the work in progress to study the case of Lagrangian submanifold $$L$$ with boundary in $$X$$ such that the intersection of $$L$$ with $$D$$ is the boundary of $$L$$.

### Sheel Ganatra

###### Structural results for wrapped Fukaya categories

I will describe, and give applications of, a series of new structural results for wrapped and partially wrapped Fukaya categories, emphasizing relationships to noncommutative geometry, mirror symmetry, and sheaf theory. This is joint work with J. Pardon and V. Shende.

### Vasiliy Golyshev

###### Dimensional interpolation and the Selberg integral

I'll show how a version of dimensional interpolation for the Riemann–Roch–Hirzebruch formalism in the case of a grassmannian leads to an expression for the Euler characteristic of line bundles in terms of a Selberg integral. I'll discuss a way to interpolate higher Bessel equations and link the result with the dimensional interpolation of the RRH formalism in the spirit of the gamma conjectures. (Joint with D. van Straten and D. Zagier.)

### Lino Grama

###### A Lie theoretical construction of a Landau–Ginzburg model and applications

In this talk we will describe the Fukaya–Seidel category of a Landau–Ginzburg model $$LG(2)$$ for the semisimple adjoint orbit of $$sl(2, \mathbb{C})$$. We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. This is a joint work with Ballico, Barmeier, Gasparim and San Martin.

### Hiroshi Iritani

###### Quantum cohomology of toric blow-ups

I will describe the change of the quantum cohomology D-modules of toric orbifolds under toric birational transformations. The analysis is based on mirror symmetry for toric orbifolds studied in joint work with Coates, Corti and Tseng. In the case of toric blow-ups, we will relate the change of quantum cohomology with that of K-groups (or derived categories) via the gamma-integral structure.

### Maxim Kontsevich

###### Birational invariants from quantum cohomology

I'll formulate a set of conjectures (some of them are within reach), on the behavior under blow-ups of Gromov–Witten invariants, Fukaya categories and semi-orthogonal decompositions. These conjectures will provide a very strong class of birational invariants, also for non-algebraically closed fields. One of relevant questions is the behavior of the Fukaya category of smooth projective variety over complex numbers, under the action of discontinuous automorphisms of $$\mathbb{C}$$. This is a joint project with L. Katzarkov and T. Pantev.

### Alexander Kuznetsov

###### Residual categories of Grassmannians

I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. As an example, I will describe residual categories of some (isotropic) Grassmannians. This is a joint work with Maxim Smirnov.

### Wanmin Liu

###### Fourier–Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic surfaces

On a Weierstrass elliptic surface $$X$$, we define a 'limit' of Bridgeland stability conditions, denoted as $$Z^l$$-stability, by varying the polarisation along a curve in the ample cone. We show that a slope stable torsion-free sheaf of positive (twisted) degree or a slope stable locally free sheaf is taken by a Fourier-Mukai transform to a $$Z^l$$-stable object, while a $$Z^l$$-semistable object of nonzero fiber degree can be modified so that its inverse Fourier-Mukai transform is a slope semistable torsion-free sheaf. As an application, on a Weierstrass elliptic surface of Picard rank two with a negative section, we show that a line bundle of fiber degree at least 2 is taken by the inverse Fourier–Mukai transform to a slope semistable locally free sheaf. This is a joint work with Jason Lo.

###### Real and symmetric matrices

I’ll describe how to implement a hyperkahler rotation relating real and symmetric matrices. Based on joint work with T.-H. Chen.

### Nikita Nekrasov

###### Lefschetz thimbles in $$O(n)$$ model and algebraic integrable systems

In the modern approach to path integrals viewed as periods the starting point is the classification of critical points of the action functional analytically continued to the complexification of the space of fields. We report on the progress in the case of the $$O(n)$$ sigma model, focusing on examples involving elliptic Calogero–Moser system and Gaudin model. Based on the joint work with I. Krichever.

### Tony Pantev

###### Relative shifted symplectic structures

This is a report on a recent joint work with Dima Arinkin and Bertrand Toen. I will discuss the notion of a relative shifted symplectic structure on a sheaf of derived stacks over a locally compact Hausdorff topological space. I will describe a general pushforward construction of relative symplectic forms and in the constructible case will explain explicit techniques for computing such forms. I will present several applications: a relative lift of recent results of Shende-Takeda on topological Fukaya categories, a universal construction of symplectic structures on derived irregular character varieties, and if time permits, a construction of shifted symplectic structures on the moduli of higher dimensional Stokes data.

### Paul Seidel

###### Fano manifolds and formal groups

Using the example of projective space blown up at an elliptic curve, I will explain how to extract arithmetic information about the mirror from Gromov–Witten invariants.

### Bernd Siebert

###### Intrinsic miror symmetry

In the talk I will sketch the completion of the joint project with Mark Gross on the intrinsic construction of mirror families for pairs $$(X,D)$$ and their degenerations, assuming only $$K_X$$ nef or $$(X,D)$$ log Calabi-Yau, as announced in our survey with the same title from 2016. The construction is based on the notion of punctured Gromov-Witten invariants under development in parallel jointly with Dan Abramovich and Qile Chen.

### Atsushi Takahashi

###### Maximally graded matrix factorizations for an invertible polynomial of chain type

In 1977, Orlik–Randell conjectured a distinguished basis of vanishing cycles with a nice combinatorial property in the Milnor fiber associated to an invertible polynomial of chain type. Motivated by their conjecture and HMS, we construct a full exceptional collection in the category of maximally graded matrix factorizations, which is a Lefschetz decomposition with respect to a polarization in the sense of Kuznetsov–Smirnov.

### Constantin Teleman

###### Coulomb branches and old-fashioned topology

I will review the construction of Coulomb branches from (3D reduced) 4D supersymmetric gauge theory with a linear representation. In the polarizable case, I will characterize that to topological boundary conditions coming from Gromov-Witten theory. The non-polarizable discussion develops ideas of Braverman, Finkelberg and Nakajima with some classical topology tweaks. The last discussion developed from conversations with Sam Raskin.

### Yukinobu Toda

###### On categorical Donaldson–Thomas theory for local surfaces

I will introduce the notion of categorical Donaldson-Thomas theories for moduli spaces of stable sheaves on the total space of a canonical line bundle on a smooth projective surface. They are constructed as certain gluings of locally defined triangulated categories of matrix factorizations, via the linear Koszul duality together with the theory of singular supports for coherent sheaves on quasi-smooth derived stacks. I will also show that the moduli stack of $$D0-D2-D6$$ bound states on the total space of a canonical line bundle is isomorphic to the dual obstruction cone over the moduli stack of pairs on the surface. This result is used to define the categorical Pandharipande-Thomas theory on the local surface. I will propose several conjectural wall-crossing formulas of categorical PT theory, motivated by d-critical analogue of D/K conjecture in birational geometry. Among them, I will show the wall-crossing formula of categorical PT theory with irreducible curve classes.

### Dmitry Tonkonog

###### Floer-theoretic heaviness and Lagrangian skeleta

I will talk on what I have learned from Umut Varolgunes, and on our work in preparation. In 2007, Entov and Polterovich introduced the notion of heavy and superheavy subsets of symplectic manifolds; such subsets have remarkable symplectic rigidity properties. One can define a variant version of (super)heaviness using relative symplectic homology studied by Varolgunes. I will explain this story, and prove that Lagrangian skeleta of symplectic divisor complements of Calabi–Yaus are superheavy in this new sense.

### Kazushi Ueda

###### Homological mirror symmetry for Milnor fibers of invertible polynomials

In the talk, I will discuss homological mirror symmetry for Milnor fibers of invertible polynomials from the point of view of homological mirror symmetry for complements of ample divisors in Calabi–Yau manifolds. If the time permits, then I will also discuss the case of abelian varieties and relation to homological mirror symmetry for varieties of general type. This is a joint work in progress with Yanki Lekili.

### Ivan Yakovlev

###### Wrapped Fukaya category of Rational homology balls

Rational homology balls are an interesting series of four-dimensional Liouville domains. Lekili and Maydanskiy proved that they all have nonvanished symplectic homology. I will calculate the Wrapped Fukaya category of these domains in two ways: using the Fukaya-Seidel category of a Lefschetz fibration and using Chekanov-Eliashberg algebra of torus knot, which have a combinatorial description.