Program
 Geometry and Physics Seminar Student No.： 50 Time： Tue 9:00-12:00, 2016-3-8~2016-5-31 Instructor： Si Li,Zhengyu Zong Place： Conference Room 3, Floor 2, Jin Chun Yuan West Building Starting Date： 2016-3-8 Ending Date： 2016-5-31

Date: 2016-7-12

Time: 10:00-12:30

Place: Conference Room 1, Floor 1, Jin Chun Yuan West Building

Speaker: Shen Linhui

Title: Donaldson-Thomas transformations for moduli spaces of local systems on surfaces.

Abstract: Kontsevich and Soibelman defined Donaldson-Thomas invariants of a 3d Calabi-Yau category with a stability condition. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in single formal automorphism of the cluster variety, called the DT-transformation.

An oriented surface S with punctures, and a finite number of special points on the boundary give rise to a moduli space, closely related to the moduli space of PGL(m)-local systems on S, which carries a canonical cluster Poisson variety structure. We determine the DT-transformation of this space.
This is a joint work with Alexander Goncharov.

Date: 2016-5-24

Time: 9:00-10:30

Speaker: Jingyue Chen (Tsinghua University)

Title: Toric GIT and Polyhedra (3)

Abstract: A full understanding of the GIT quotients introduced last time involves the interesting polyhedra associated to characters. We will discuss the polyhedron of a character and then show that the GIT quotient is a semiprojective toric variety when G is an algebraic subgroup of (C*)^r.

Time: 10:30-12:00

Speaker: Zhilan Wang (Academy of Mathematics and Systems Science)

Title: Hilbert schemes, Quot schemes and moduli spaces of surfaces

Abstract: In 1999, Lehn proposed a conjecture on the closed formula for the integrals of the top Segre classes of tautological bundles on Hilbert schemes of points on surfaces. In this talk I will review how Marian, Oprea and Pandharipande proved Lehn's conjecture for K3 surfaces X using localization on the Quot schemes of X. Moreover, they obtained new systems of relations among tautological classes on moduli spaces of surfaces and their relative Hilbert schemes of points via localization.

Reference: Marian A, Oprea D, Pandharipande R. Segre classes and Hilbert schemes of points. arXiv preprint arXiv:1507.00688 (2015)

Date: 2016-5-17

Time: 9:00-10:30

Speaker: Jingyue Chen (Tsinghua University)

Title: Toric GIT and Polyhedra (2)

Abstract: A full understanding of the GIT quotients introduced last time involves the interesting polyhedra associated to characters. We will discuss the polyhedron of a character and then show that the GIT quotient is a semiprojective toric variety when G is an algebraic subgroup of (C*)^r.

Time: 10:30-12:00

Speaker: Shanzhong Sun (Capital Normal University)

Title: Gutzwiller's semiclassical trace formula and Maslov index

Abstract: We will recall the physical derivation of the semiclassical trace formula due to M. Gutzwiller which relates the energy spectrum of a quantum mechanical system to the corresponding chaotic classical Hamiltonian system, in particular its closed orbits, and clarify the role of the Maslov index in this formula. Some possible future investigations will be discussed.

Date: 2016-5-10

Time: 9:00-10:30

Speaker: Jingyue Chen (Tsinghua University)

Title: Toric GIT and Polyhedra

Abstract: A full understanding of the GIT quotients introduced last time involves the interesting polyhedra associated to characters. We will discuss the polyhedron of a character and then show that the GIT quotient is a semiprojective toric variety when G is an algebraic subgroup of (C*)^r.

Time: 10:30-12:00

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity 5

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Date: 2016-4-26

Time: 10:30-12:00

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity 4

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Date: 2016-4-19

Time: 9:00-10:30

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity (3)

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Time: 10:30-12:00

Speaker: Yi Huang (Tsinghua University)

Title：Moduli space volumes and Witten's conjecture

Abstract：We consider the problem of computing the Weil-Petersson volume of the moduli space of curves, and demonstrate Mirzakhani's solution to this problem. In so doing, we motivate how these volume computations relate to intersection numbers on (Delign-Mumford compactified) moduli spaces and how Mirzakhani's work led to a proof of Witten's conjecture.

Date: 2016-4-12

Time: 9:00-10:30

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Time: 10:30-12:00

Speaker: Hao Wen (Peking University)

Title: A twisted Cauchy-Riemann operator: local and global

Abstract: Given a quasi-homogeneous polynomial f, we define a twisted Cauchy-Riemann operator as \bar\partial_f=\bar\partial+\partial f\wedge. We will study its L^2 theory on \mathbb{C}^n and on a smooth bounded strongly pseudoconvex domain. Related objects are tt^* equations and Witten deformation.

Date: 2016-4-5

Time: 9:00-10:30

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Time: 10:30-12:00

Speaker: Zhe Sun (Tsinghua University)

Title: Rank n swapping algebra for the Hitchin component, its quantization and TQFT

Abstract:I will introduce the rank n swapping algebra which characterize the Hitchin component. This algebra is also related to local functions of Fock-Goncharov's X variety. I will also introduce a deformation quantization of this algebra. This quantization covers the quantization of Fock-Goncharov's quantiztion of PGL(n,R) X variety. If possible, we will also discuss TQFT related to rank n swapping algebra.

Date: 2016-3-29

Time: 9:00-10:30

Speaker: Weiqiang He (Tsinghua University)

Title: Global B-model for simple elliptic singularity

Abstract: Global B-model is an important tools in cohomological field theory. It is very useful in many topic, such as LG-CY correspondence, modularity and crepant resolution correspondence. In this reading seminar, I will show how to construct global B-model for simple elliptic singularity f=x^3+y^3+z^3, this is a work by T. Milanov and Y. Ruan.

Ref: T. Milanov, Y. Ruan Gromov-Witten theory of elliptic curve and quasi-modular forms arXiv:1106.2321

Time: 10:30-12:00

Speaker: Yin Tian (Tsinghua University)

Title: Discussion on Khovanov-Seidel's paper "Quivers, Floer cohomology, and braid group actions"

Date: 2016-3-22

Time: 9:00-10:30

Speaker: Yifan Li (Tsinghua University)

TitleWitten-Morse Theory on Real Manifolds

Abstract: In 1982[1], E.Witten introduced a relationship between supersymmetry quantum mechanics and Morse theory. He used a deformation (which was called the Witten deformation) to obtain the Morse inequality and to show the relation between supersymmetry index and Euler number. Soon after that, K.Fukaya[2] introduced the Morse category as a realization of A_\infty category and conjectured that the A_\infty product in the Morse category can be seen as a leading term of the A_\infty product of the de Rham category twisted by Witten deformation, which was proved bu Z.Ma, N.Leung, K.Chan[4]. I will introduce the development of Witten-Morse theory and recent progress in the association of geodesic trees and scattering diagrams by Z.Ma and N.Leung. This is a reading report.

Ref:

[1]E.Witten, Supersymmetry and Morse Theory, 1982

[2]K.Fukaya, Morse Homotopy A^{infty} Category and Floer Homologies, 1993

[3]W. Zhang, Lectures on Chern-Weil Theory and Witten Deformations, 2001

[4]Z.Ma, N.Leung, K.Chan, Witten Deformation of Product Structure on de Rham Complex, 2016

Time: 10:30-12:00

Speaker: Yin Tian (Tsinghua University)

Title: A categorification of a Clifford algebra via 3-dimensional contact topology

Abstract: Categorification, initiated in the work of Crane and Frenkel, describes richer higher-level" structures which aim to clarify objects on the decategorification level and to provide finer invariants. A celebrated example is Khovanov homology which categorifies the Jones polynomial. In this talk we describe a categorifcation of a Clifford algebra. The motivation is from the contact category C(D^2) of a disk D^2 introduced by Honda which describe contact structures on D^2 \times [0,1].

Date: 2016-3-15

Time: 9:00-10:30

Speaker: XinXing Tang (Peking University)

Title: Introduction to the Riemann-Hilbert problem

Abstract: To find the simplest way to present a differential system up to meromorphic equivalence is the subject of the Riemann-Hilbert problem(in the case of regular singularities) or of Birkhoff's problem. In this talk, I will introduce some basic definitions about regular singularity, logarithmic lattice, state the Riemann-Hilbert problem and some equivalent descriptions, and give the criterion that the Riemann-Hilbert problem has a solution.

Time: 10:30-12:00

Speaker: Xiaowen Hu (Peking University)

Title: Classical and quantum McKay correspondence (2)

Abstract: First I will give an introduction to the classical McKay correspondence, and explain this via Chen-Ruan cohomology. Finally I will talk about the quantum McKay correspondence.

Date: 2016-3-8

Time: 9:00-10:30

Speaker: Jingyue Chen (Tsinghua University)

Title: Introduction to toric GIT

Abstract: We will give an introduction to the geometric invariant theory of a closed subgroup G of (C^*)^r acting on C^r, which uses a character of G to lift the G-action to a trivial line bundle over C^r.

Time: 10:30-12:00

Speaker: Xiaowen Hu (Peking University)

Title: Classical and quantum McKay correspondence

Abstract: First I will give an introduction to the classical McKay correspondence, and explain this via Chen-Ruan cohomology. Finally I will talk about the quantum McKay correspondence.