### D-brane categories and mirror symmetry (online seminar )

TIME:May11- July 8 ，2020 (Every Wednesday,16:00-18:00 Beijing time)
LOCATION:Zoom online conference
##### Description

This workshop contains several series talks and some lectures. It will concentrate on topics in gauged linear sigma models, derived categories of coherent sheaves, Fukaya categories and the homological mirror symmetry conjecture. The workshop intends to bring experts in the ﬁeld to exchange ideas, and also to help graduate students to learn various categorical theories and dualities between them.

The arrangement of the first three weeks is as follows:
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Lecture series I for 1st week, May 11-15, Prof. Chris Brav, HSE, Moscow

1) Intro to derived algebraic geometry and deformation theory; Time: 16:00 pm (Beijing time), May 11th, Monday.

Time: 16:00 pm (Beijing time), May 11th, Wednsday.

2) Examples of deformations of schemes (and more general prestacks), derived stabiliser groups, deformations of Calabi-Yau varieties, and a 'proof from the book’ of the Bogomolov-Tian-Todorov lemma

Time: 16:00 pm (Beijing time), May 13th, Wednsday.

ExpirationTime：2021-07-31 23:59

3) Deformations of dg categories and of Calabi-Yau dg categories.

Time: 16:00 pm (Beijing time), May 15th, Friday.

ExpirationTime：2021-07-31 23:59

Lecture series II for 2nd week, May 18-22, Prof. Johnna Knapp, Melbourne.

1) Gauged linear sigma model I

Time: 16:00 pm (Beijing time), May 18th, Monday.

ExpirationTime：2021-07-31 23:59

2) Gauged linear sigma model II

Time: 16:00 pm (Beijing time), May 20th, Wednsday.

ExpirationTime：2021-07-31 23:59

3) Gauged linear sigma model III

Time: 16:00 pm (Beijing time), May 22th, Friday.

ExpirationTime：2021-07-31 23:59

Lecture series III  for 3rd week, May 25-29

1) Non-commutative  primitive forms of matrix factorizations, I

Speaker: Prof. Junwu Tu, ShanghaiTech University

Time: 15:50 pm (Beijing time), May 25th, Monday.

Abstract:

In this talk, we present an introduction to Non-commutative Hodge theory.

We shall sketch constructions of the main ingredients in the theory, such as Hochschild Homology, cyclic homology, Mukai pairing, Getzler’s connection, and so on.

To join VooV Conference:

Conference ID：957 046 487

2) Non-commutative primitive forms of matrix factorizations, II

Speaker: Prof. Junwu Tu, ShanghaiTech University

Time:15:50 pm (Beijing time), May 27th, Wednsday.

Abstract:

In this talk, we study the Non-commutative Hodge theory in the case of matrix factorizations.This naturally leads to a categorical interpretation of K. Saito’s theory of primitive forms. This allows us to generalize Saito’s theory to the orbifold case.

To join VooV Conference:

Conference ID：599 081 473

3)Topic: Non-commutative primitive forms and Gromov-Witten invariants

Speaker:Prof. Lino Amorim (Kansas State University)

Time:15:00 PM (Beijing time), May 29th, Friday

Abstract:

I will describe a non-commutative analogue of Saito's theory of primitive forms. Under some conditions, this construction associates to a Calabi-Yau A-infinity category a Frobenius manifolds over its Hochschild homology. When applied to the Fukaya category, it is expected this construction recovers the (genus zero) Gromov-Witten invariants of the symplectic manifold. I’ll show this is indeed the case for most Fano toric manifolds, in particular projective spaces. This is joint work with Junwu Tu.

To join ZOOM Conference:

RECORDING:https://zoom.us/rec/share/4MJtdZ7M-HxJWbOc5hiFWoUjBoDvX6a80HBPr_cPmkxQ-XhCyMDMhRFCSiAjdylc

Topic:The Deligne-Illusie method for Artin stacks

Speaker:Artem Prikhodko (Higher School of Economics)

Time:16:00 PM (Beijing time), Jun 3

Abstract: Deligne-Illusie's method establishes degeneration of the Hodge-to-de Rham spectral sequence for smooth proper schemes in characteristic 0 by reduction to positive characteristic. In the talk, I will explain how to adapt this approach to Artin stacks. As an application we deduce equivariant Hodge-to-de Rham degeneration for a few families of varieties with algebraic group action. This is joint work with Dmitry Kubrak.

To join the zoom conference :

https://zoom.com.cn/j/66039310426?pwd=YWlXMkZpeDh3ODFCOHQ2dE5LSExidz09

ID：660 3931 0426

Topic: Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

Speaker:Alexander Efimov（Steklov Institute/Laboratory for Mirror Symmetry and Automorphic Forms, HSE）

Time:16:00 PM (Beijing time), Jun 10

Abstract:

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin). In particular, we show that there exists a minimal $10$-dimensional $A_{\infty}$-algebra over a field of characteristic zero, for which the supertrace of $\mu_3$ on the second argument is non-zero.

As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification, giving a negative answer to a question of To\"en.

This can be interpreted as a lack of resolution of singularities in the noncommutative setup.We also obtain an example of a proper DG category which does not admit a categorical resolution of singularities in the terminology of Kuznetsov and Lunts (that is, it cannot be embedded into a smooth and proper DG category).

NOTE：only need to paste the conference address to the browser to join the conference

Valid Until: 2025-07-31 23:59

Topic:Computing Symplectic Cohomology via Mirror Symmetry

Speaker:Yanki Lekili (King's College London)

Time:16:00 PM (Beijing time), Jun 17

Abstract: Morse-Bott methods for determining symplectic cohomology lead to limited success. I will describe several cases, including Milnor fibers of Kleinian singularities, where we can prove homological mirror symmetry and use this to explicitly compute symplectic cohomology by algebro-geometric methods. This roundabout method ultimately leads to an easy algorithmic computation of symplectic cohomology for Milnor fibers of a general class of quasihomogeneous hypersurface singularities. The talk is based on joint work with Kazushi Ueda.

To join the Conference:
https://bbb.freemath.xyz/b/yan-rat-dhm

Speaker:  Prof. Emanuel Scheidegger（ BICMR,PKU）
Time：16:00-18:00 （Beijing Time）  June 24 ，2020
Title： D-brane central charge and Landau-Ginzburg orbifolds

Abstract:The D-brane central charge is a function on the the K-theory of a triangulated category and plays an essential role in Bridgeland stability. We are interested in the situation in which triangulated categories of different origin are related by a correspondence. The prominent example is Orlov's correspondence between a category of matrix factorizations coming from a Landau-Ginzburg orbifold and the derived category of coherent sheaves on the corresponding Calabi-Yau manifold. This situation has a nice description in physics in terms of the gauged linear sigma model. We propose a conjecture for an explicit formula for the D-brane central charge in the various regions of the Kaehler moduli space. We verify this conjecture for a large class of Landau-Ginzburg orbifolds.

To join the Zoom conference:
https://zoom.com.cn/j/4162454234?pwd=cThmTE1LUHVUanU5dTlYTHNMZUMvUT09

ID：416 245 4234

Speaker:  Prof. Emanuel Scheidegger（ BICMR,PKU）
Time：16:00-18:00 （Beijing Time）  July 01 ，2020
Title： D-brane central charge and Landau-Ginzburg orbifolds II

Abstract:Last time we learned about the D-brane central charge in mathematics and physics, and we suggested to view it as a function on the stringy Kaehler moduli space of a variation of GIT quotients and their associated triangulated categories. This situation has a nice description in physics in terms of the gauged linear sigma model whose phases correspond to various regions of the Kaehler moduli space. This time we focus on the gauged linear sigma model and on the Landau-Ginzburg orbifold phase to compare the resulting D-brane central charge functions.
To join the Zoom conference:
https://zoom.com.cn/j/4162454234?pwd=cThmTE1LUHVUanU5dTlYTHNMZUMvUT09
ID：416 245 4234

Speaker:  Prof. Alimjon Eshmatov（ University of Toledo）

Time：17:00-19:00 （Beijing Time）  July 08 ，2020
Title： Perverse sheaves and knot contact homology
Abstract:A knot is an oriented closed connected smooth curve in R3 . A general curve with possibly many components is called a link. One of the fundamental problems in topology is to classify knots by suitable invariants. In this talk we present a universal construction, called homotopy braid closure, that produces invariants of links in R3 starting with a braid group action on objects of a (model) category. Applying this construction to the natural action of the braid group Bn on the category of perverse sheaves on the two-dimensional disk with singularities at n marked points, we obtain a differential graded (DG) category that gives knot contact homology in the sense of L. Ng. As an application, we show that the category of finite-dimensional modules over the 0-th homology of this DG category is equivalent to the category of perverse sheaves on R3 with singularities at most along the link. [This is joint work with Yu. Berest and
Wai-kit Yeung

Video：

Valid Until: 2025-08-31 23:59

##### Organization Committee

◆ Christopher BRAV (High School of Economics)

◆ Huijun FAN (School of Mathematical Sciences, Peking University)

◆ Emanuel SCHEIDEGGER (Beijing International Center for Mathematical Research, Peking University)

◆ Junwu TU (Shanghai Tech University)

##### Contact

Room 1275, Science Building No .1, Peking University, Beijing, China 100871

Phone: +86 10 62744768

E-mail:zhangxy@math.pku.edu.cn

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