# Sino-Russian Hybrid Conference “Geometry and Physics”

**• Speakers:**

**Fedor Bogomolov** (NYU, USA, NRU HSE, Russia), *Extensions of stable vector bundles on projective manifolds and Inequalities for Chern classes (joint work with E.Lukzen and V. Zhgoon). *

I will discuss an approach to obtain from stability conditions for vector bundle extensions classical BMY inqualities for Chern classes of projective manifolds. I also discuss their generalizations in the presence of higher dimensional space of formal deformations. I will also touch some related problems in geometry.

**Аlexey Bondal**(Kavli Institute, Japan, Steklov Mathematical Institute of RAS, MIPT, Russia), Co

*mplex geometry via twist closed enhancements. ♦*

**Valery Gritsenko** (Lille University, France, NRU HSE, Russia), * Automorphic asymmetry and moduli spaces. *

Borcherds аutomorphic products often have fundamentally different behaviour in the neighbourhoods of the boundary components of a fundamental domain. We show how to use such automorphic asymmetry to solve difficult questions in the theory of moduli spaces of Enriques surfaces (Kondo's problem) and polarised generalised Kummer manifolds.

**Ludmil Katzarkov** (Miami University, USA, NRU HSE, Russia), * A parallel reality looks at generalized geometry. *

We begin by recalling classical works of Donaldson and Simpson. Based on these works we take a new look at generalized geometries and "Hodge theory"connected with them.

**Maxim Kontsevich**(IHES, France),

*On the atomic decomposition from quantum multiplication and its applications.*

In a talk given at HSE in 2019, I proposed a way to apply genus zero Gromov-Witten invariants to questions of rationality of algebraic varieties. The main idea is based on a loose analogy with the morsification of an isolated critical point of a holomorphic function. The cohomology of any smooth projective variety defined over a field of characteristic zero (not necessarily algebraically closed) splits non-canonically into the sum of so-called atoms, whose isomorphism classes as noncommutative motives are canonical. Recently Hiroshi Iritani proved so-called blowup formula for quantum cohomology, which is the key result for the program, and describes the structure of atoms for the blowup. Based on Iritani's result one can conclude now e.g. that the very general cubic 4-fold is not rational. I'll give a review of new invariants, and propose several conjectures refining the structure of atoms and related to Bridgeland stability structures on derived categories of coherent sheaves. The talk is based on the ongoing project with L. Katzarkov, T. Pantev and T. Yu, as well as a related another project with D. Auroux and L. Katzarkov.

**Aleksandr Kuznetsov** (Steklov Mathematical Institute of RAS, Russia), * One-nodal degenerations of Fano threefolds. *

There are two sorts of 1-nodal degenerations of Fano 3-folds, factorial and non-factorial. I will describe a classification of such degenerations for 3-folds of Picard rank 1, emphasizing an interesting relation with smooth Fano 3-folds of higher Picard rank. This is a joint work in progress with Yu. Prokhorov.

**Andrei Losev** (NRU HSE, Russia), *Tropical mirror. *

In this talk I will discuss tropical mirror developed by me and S.Lysov in three papers: arXiv:2204.06896, arXiv:2301.01687, arXiv:2305.00423 based on the concept of Higher Topological Quantum Mechanics (HTQM) on graphs (arXiv:2112.12756). HTQM is a collection of data in the linear algebra and I will start with explaining it. Then I will briefly explain how HTQM may be used in proving Kadeishvili theorems for A-infinity and L-infinity case (see also arXiv:2112.12756). I will proceed with explanation how problem of counting of tropical curves passing through tropical cycles may be understood as HTQM with circle action –we will call it A-model. Then sum over graphs in A-model would be transformed into sum over graphs in another HTQM -B-model (of BCOV type). In this process differential in B-model will happen to be differential in LG Quantum mechanics, and evaluation operators would be transformed into mirror states. These states would form a good section in terms of K. Saito theory with exponential superpotential.

**Dmitri Orlov** (Steklov Mathematical Institute of RAS, Russia), * Smooth DG algebras and twisted tensor product. ♦*

**Nikolai Reshetikhin** (YMSC, Tsinghua University, BIMSA, China), * Integrable systems on moduli spaces of flat connections over surfaces. *

The talk will start with the description of integrable (superintegrable) classical Hamiltonian systems on moduli spaces of flat connections over surfaces. In the second part of the talk corresponding quantum integrable systems will be described. When the underlying Lie group is compact simple, level surfaces of integrals in the classical case are compact and quantum spaces for corresponding quantum systems are finite dimensional.

**Peng Shan** (YMSC, Tsinghua University, China), * Modularity for W-algebras and affine Springer fibers.*

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibers. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.

**Viacheslav Shokurov** (Johns Hopkins University, USA, Steklov Mathematical Institute of RAS, Russia), *3D birational geometry and special classes of conic bundles. *

Two classes of 3-fold conic bundles will be introduced. It will be explained their role 3-dimensional birational geometry.

**Mauricio Romo** (Tsinghua University, China), *Gauged linear sigma models for mirrors of singular double covers and their noncommutative resolutions. *

I will present a construction of mirrors of singular Calabi-Yau double covers via noncommutative resolutions realized by gauged linear sigma models with hybrid phases. I will focus on the correspondence between A and B-periods and present some conjectures about their derived categories and other invariants. This is based on joint work with Bong Lian and Tsung-Ju Lee.

**Jinxin Xue** (Department of Mathematics, Tsinghua University, China), *Global dynamics of the N-body problem. *

The N-body problem is a fundamental model in classical mechanics. It continues to play an important role in modern physics and mathematics due to its simplicity and richness of dynamical behaviors such as the existence of chaos, noncollision singularities, etc. In this talk, we give an overview of the dynamics of the N-body problem and explain our work on the existence of noncollision singularities and superhyperbolic orbits.

**Xiaomeng Xu (** Beijing University, China), * Crystals arising from the WKB analysis. *

This talk gives an introduction to the Stokes phenomenon of an irregular Knizhnik–Zamolodchikov equation, associated to a representation L(\lambda) of gl_n. It then proves that the Stokes matrices of the equation produce representations of quantum group U_q(gl_n) on L(\lambda). Motivated by the relation with quantum group, it further gives a realization of the gl_n-crystal structures on L(\lambda) via the WKB approximation of the Stokes matrice.

**Youjin Zhang** (Department of Mathematics, Tsinghua University, China), *Generalized Frobenius manifolds with non-flat unity and bihamiltonian integrable hierarchies. *

We consider the relation between a class of generalized Frobenius manifolds with bihamiltonian integrable hierarchies. Such generalized Frobenius manifolds are defined by Dubrovin’s definition of Frobenius manifold but without the flatness condition imposed on the unit vector fields. Important examples arise from the study of Gromov-Witten theory and Hodge integrals. For any semisimple generalized Frobenius manifold, we present an analogue of the construction of bihamiltonian integrable hierarchies that was given by Dubrovin and Zhang for semisimple Frobenius manifolds.

**Shing-Tung Yau** (YMSC, Tsinghua University), * Canonical Hermitian Metrics and non-KahlerCalabi-Yau Threefolds . *

A conifold transition is a process which degenerates 2-cycles and introduces new 3-cycles. We will investigate the web of Calabi-Yau threefolds connected by conifold transitions. Early studies on conifold transitions includes works of Reid and Friedman in the algebraic geometry literature, and Candelas-de la Ossa and Candelas-Green-Hubsch in the physics literature. A conifold transition may connect a Kahler Calabi-Yau to a non-Kahler complex manifold with trivial canonical bundle. There is a also a reverse conifold transition which may produce a non-Kahler symplectic manifold. In this talk, we will discuss the geometrization of these spaces by canonical metrics.

**• Schedule :August 28, 2023 - online August 29, 2023 - online August 30, 2023 - Usacheva Street Bld. 6 August 31, 2023 - Gubkina Street Bld. 8 **

**•**Organizers: Valery Gritsenko, Sergey Gorchinskiy, Nikolai Reshetikhin.

**•**Scientific committee: Aleksandr Belavin, Ludmil Katzarkov, Dmitri Orlov, Shing-Tung Yau.

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