Matrix Factorizations and Mirror Symmetry
2018.10.15-19 KIAS 8101 |

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**Hulya Arguz**

TALK I:

Title: The Gross-Siebert program in mirror symmetry

Abstract: An algebro-geometric approach to the Strominger-Yau-Zaslow conjecture has been developed by Gross and Siebert. In this talk we will review the main tools used in this approach. In particular, we will discuss toric degenerations constructed from integral affine manifolds with singularities and log geometric techniques used in the construction of such degenerations.

TALK II:

Title: Real Lagrangians in toric degenerations

Abstract: Real loci of toric degenerations provide an ample source of examples of Lagrangians, which are amenable to Floer theoretic tools. In this talk, we will describe how the topology of these Lagrangians can be understood by means of affine and log geometry. We then will restrict ourselves to the quintic threefold and explain how the topology of such Lagrangians relates to the Hodge numbers of the quintic. The first part of this talk is based on joint work with Bernd Siebert and the second part is joint work with Thomas Prince.

**Michael Brown**

Title: Topological K-theory of matrix factorization categories

Abstract: This is joint work with Tobias Dyckerhoff. Topological K-theory of complex-linear dg categories is a notion introduced by Blanc in his paper "Topological K-theory of complex noncommutative spaces". In these talks, I will discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a quasi-homogeneous polynomial with complex coefficients in terms of a classical topological invariant of a complex hypersurface singularity: the Milnor fiber and its monodromy. Along the way, I will provide background on Blanc's notion of topological K-theory for dg categories.

**Cheol-hyun Cho **

Title: Gluing Localized mirror functors

Abstract: We develop a method of gluing localized mirrors and functors from immersed Lagrangians in the same deformation class. In the first lecture, we give detailed algebraic construction leading to the gluing theorem where a single criterion is used to glue mirror space, as well as functors. In the second lecture, we discuss the example of punctured Riemann surfaces and its homological mirror symmetry.

**Yunhyung Cho **

Title: Monotone Lagrangians in flag varieties

Abstract: A flag variety is a complex homogeneous space and is an important example of a smooth Fano variety. Any flag variety X of type A, B, or D is equipped with a completely integrable system(called a Gelfand-Cetlin system) whose image is a convex polytope. In case of type A, Nishinou-Nohara-Ueda calculated the disc potentials for regular (torus) fibers and pointed out that, in general, regular fibers are not enough to generate the Fukaya category by the appearance of Lagrangian fibers over the boundary. In this talk, we describe non-torus Lagrangian fibers explicitly and explain how to find some of those who can possibly give a non-zero object in the Fukaya category. This is joint work with Yoosik Kim and Yong-Geun Oh.

**Yuki Hirano **

Title: Derived factorization categories of non-Thom--Sebastiani-type sum of potentials.

Abstract: We give a semi-orthogonal decomposition of derived factorization categories associated to sum of certain potentials that is not Thom--Sebastiani-type. As an application, we prove that the homotopy category of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type has a full exceptional collection whose length is the Milnor number of the Berglund–Hu Ìˆbsch transpose of $f$. This is joint work with Genki Ouchi.

**Atsushi Kanazawa**

Title: Gluing Landau-Ginzburg models via Tyurin degeneration

Abstract: We consider a Tyurin degeneration of a CY manifold X to a union of two quasi-Fano manifolds X1 and X2. The Doran-Harder-Thonpson conjecture claims that we should be able to glue together the mirror Landau–Ginzburg models of X1 and X2 to obtain the mirror CY manifold of X. I will explain a proof of this conjecture in the simplest case, an elliptic curve, via SYZ mirror symmetry. Then I will propose a new construction of Landau–Ginzburg models inspired by this conjecture.

**Sangwook Lee **

Title: Pairing structures in mirror symmetry between symplectic manifolds and Landau-Ginzburg B-models

Abstract: We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. Such a relation follows from multi-crescent Cardy identity, which is a generalized form of Cardy identity. There is an interesting conformal factor between these two pairings, which can be described as a ratio of Floer volume class and classical volume class. We also introduce a new kind of invariants of Lagrangian Floer cohomology with values in Jacobian ring of the mirror potential function. As an application, we discuss the case of general toric manifold, and the relation to the work of Fukaya-Oh-Ohta-Ono and their Z-invariant. Also, we compute the conformal factor for an elliptic curve quotient which is expected to be related to the choice of a primitive form. This is a joint work with Cheol-hyun Cho and Hyung-Seok Shin.

**Yuuki Shiraishi**

Title: On Frobenius structures from cusp singularities

Abstract: There is a semi--infinite Hodge structure called Saito structure associated to deformation theory of an arbitrary isolated hypersurface singularity. In this talk, we determine primitive forms (at origin of the deformation spaces of singularities), special volume forms, for the Saito structures associated to cusp singularities. This volume forms play important roles for determining the Frobenius structures, period mappings and so on. We also show the classical mirror symmetry between the Saito theory for cusp singularities with primitive forms above and the orbifold GW theory for orbifold projective lines with negative Euler numbers. This is joint work with Atsushi Takahashi.

**Atsushi Takahashi**

Title: Mirror symmetry and the category of matrix factorizations for an invertible polymonial of chain type

Abstract: An introduction to the mirror symmetry for invertible polynomials is given. For an invertible polymonial of chain type, we propose a conjecture on the existence of a particular full exceptional collection in the maximally-graded category of matrix factorizations motivated by Orlik-Randell's conjecture in singularity theory.

**Dmitry Tonkonog **

Title: Disk potentials and mirror symmetry I, II

Abstract:It is well known that the mirror to a Fano manifold should be a Landau-Ginzburg model. It is also known, at least a guiding principle, that Landau-Ginzburg mirrors can be constructed using holomorphic disk counts on Lagrangian submanifolds. I will explain how to approach and prove several classical mirror symmetry predictions purely within the holomorphic disk viewpoint on LG models. In particular, I will discuss the mirror computation of certain GW invariants using period integrals, and the quantum Lefschetz formula. The last part is joint work in progress with Diogo, Vianna and Wu.