Date (JST):	Jun 20, 2023 - Jun 21, 2023,
Title:	Current trends in categorically approach to algebraic and symplectic geometry 2
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Remarks:	Current trends in categorically approach to algebraic and symplectic geometry 2 Date:June20-21, 2023 Venue:Kavli IPMU Lecture Hall Organizers: Alexey Bondal (Steklov Mathematical Institute) Mikhail Kapranov (Kavli IPMU) Speakers: Agnieszka Bodzenta (Warsaw Univrsity) Alexey Bondal (Steklov Math Institute) Will Donovan (Tsinghua University) Norihiro Hanihara (Kavli IPMU) Osamu Iyama (Tokyo University) Tatsuki Kuwagaki (Kyoto University) Shinnosuke Okawa (Osaka University) Yukinobu Toda (Kavli IPMU) Schedule: Tuesday 20 June 2023 10:00-11:00 W. Donovan (Tsinghua University) Monodromy for the Appell hypergeometric system and mirror symmetry 11:30-12:30 A. Bodzenta (Warsaw University) Periodic SODs for root stacks 14:00-15:00 Sh. Okawa (Osaka University) The infinite dihedral group and noncommutative quadrics 16:00-17:00 Yu. Toda (Kavli IPMU) Quasi-BPS categories for K3 surfaces 17:00-18:00 Free discussion (seminar room 4-th floor) Wednesday 21 June 2023 10:00-11:00 T. Kuwagaki (Kyoto University) Curved dga for Lagrangian submanifold from sheaf theory 11:30-12:30 A. Bondal (Steklov Math Institute) Noncommutative resolutions and their null categories 14:00-15:00 N. Hanihara (Kavli IPMU) Non-commutative resolutions for Segre products 16:00-17:00 O. Iyama (Tokyo University) Singularity categories and cluster categories 17:00-18:00 Free discussion (seminar room 4-th floor) Abstracts: Agenieszka Bodzenta (Warsaw University) Periodic SODs for root stacks I will recall two constructions of the r'th root stack associated to a section of a line bundle. One of the constructions will allow me to view the root stack from the variation of GIT point of view. I will describe the associated semi-orthogonal decompositions and argue that they are 2r periodic. In particular, for r=2, they give a spherical functor. Finally, I will give a possible application in the case of flops. This is a report on a joint work in progress with Will Donovan. Alexey Bondal (Steklov Math. Institute) Noncommutative resolutions and their null categories We discuss Auslander noncommutative resolutions of singularities of finite MCM type and their null categories. Knoerrer periodicity for these null categories holds. There are spherical functors related to these resolutions and derived equivalences of partial Auslander resolutions. Will Donovan (Tsinghua University) Monodromy for the Appell hypergeometric system and mirror symmetry Monodromy for the Gauss hypergeometric system corresponds to an action of derived symmetries on K-theory of the resolved conifold, namely a resolution of the 3-fold quadric cone. Multi-parameter generalizations of the Gauss system were first studied by Appell and Lauricella in the late 19th century. I will describe how monodromy for these multi-parameter systems also corresponds to an action of derived symmetries, with the resolved conifold replaced with certain higher-dimensional cones of tensors. This work grew from discussions with Tatsuki Kuwagaki, and is joint with Weilin Su. Norihiro Hanihara (Kavli IPMU) Non-commutative resolutions for Segre products Abstract: Non-commutative resolutions serve as models of resolutions of singularities via non-commutative rings, and at the same time as natural domains for higher dimensional Auslander-Reiten theory. They are therefore interesting objects to study both in algebraic geometry and representation theory. In the talk we will explain a construction of non-commutative resolutions and its application to representation theory. Osamu Iyama (Tokyo University) Singularity categories and cluster categories I will compare two well-studied classes of representations: quiver representations and Cohen-Macaulay representations. The representation-finite algebras in both classes are parametrized by Dynkin diagrams by theorems due to Gabriel and Buchweitz-Greuel-Schreyer. Both classes give rise to Calabi-Yau triangulated categories: cluster categories and singularity categories. I will explain that there are a number of triangle equivalences between cluster categories and singularity categories. This is a joint work with Norihiro Hanihara (arXiv:2209.14090). Tatsuki Kuwagaki (Kyoto University) Curved dga for Lagrangian submanifold from sheaf theory For a Lagrangian brane, Floer theory associates a curved A_\infty algebra. Each Maurer—Cartan element of this algebra gives an object of non-curved Fukaya category. In this talk, I'll explain the sheaf-theoretic counterpart of this story. Shinnosuke Okawa (Osaka University) The infinite dihedral group and noncommutative quadrics It is well known that the set of markings of a del Pezzo surface is a torsor under an action of the Weyl group of an appropriate type. For noncommutative del Pezzo surfaces we expect something similar where the Weyl group is replaced with the affine Weyl group. I will explain how this works in the case of noncommutative quadrics (namely, noncommutative deformations of $\mathbb{P}^1 \times \mathbb{P}^1$). The main result is that we can reconstruct the 3-dimensional Artin-Schelter regular cubic $\mathbb{Z}$-algebra from the associated noncommutative quadric exactly up to a certain action of the infinite dihedral group, that is the affine Weyl group of type A1. Yukinobu Toda (Kavli IPMU) Quasi-BPS categories for K3 surfaces In this talk, I will give semiorthogonal decompositions of derived categories of coherent sheaves on moduli stacks of semistable objects on K3 surfaces. An each summand is given by the categorical Hall product of subcategories called quasi-BPS categories, which approximate the categorification of BPS cohomologies for K3 surfaces. When the weight and the Mukai vector is coprime, the quasi-BPS category is shown to be smooth and proper, with trivial Serre functor etale locally on the good moduli space. So it gives a twisted analogue of categorical crepant resolution of the singular symplectic moduli space, and reminiscents categorical analogue of chi-independence phenomena. This is a joint work in progress with Tudor Padurariu. Organizers: Alexey Bondal (Steklov Math Institute), Mikhail Kapranov (Kavli IPMU) Sponsor/Cosponsor: Kakenhi Kavli IPMU

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