## Welcome to GAP seminar

This seminar aims at filling up the gap between graduate level math and research math, and enlarging the scope of graduate students as well as more advanced researchers. In each talk, the speaker will focus on a currently active research topic. He/she will spend one hour or so to introduce the background of the topic, including basic conceptions/examples, important known results and major problems etc. The last part of the talk will usually be more technical and is related to the speaker's own work. The subjects of future talks are mainly chosen from geometry(G), algebra, analysis(A), and mathematical physics(P).

## Upcoming Talks

## Past Talks

Archive of GAP Seminar: Fall 2013 | Spring2014 | Fall 2014 | Spring 2015 | Fall 2015 | Spring 2016 | Fall 2016 | Spring 2017 | Fall 2017

Date: 6/15, 16:00-17:30

Speaker: Weiyi Zhang（University of Warwick）

Title: From smooth to almost complex

Abstract: An almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex or symplectic manifold is an almost complex manifold, but not vice versa.

Transversality is the notion of general position in manifold topology. It could be used to define the multiplicity of a zero for a smooth function with more than one variables. We will discuss differential topology of almost complex manifolds, explain how to use transversality statements for smooth manifolds to formulate and prove corresponding results for an arbitrary almost complex manifold. The examples include intersection of almost complex manifolds, pseudoholomorphic maps and zero locus of certain harmonic forms. Using these results, we are able to define and study Kodaira dimension for almost complex manifolds (joint with Haojie Chen). It could also be applied to get upper bounds for eigenvalues of Laplacian on almost complex manifolds (work of my student Louis Bonthrone).

An undergraduate student who has taken Prof. Zuoqin Wang's Manifold class should be able to understand a large portion of the talk.

Date: 5/30, 10:00-11:30

Speaker: Huai-Liang Zhang（Hong Kong University of Science and Technology）

Title: MSP theory: counting curves in Quintic Calabi Yau threefolds

Abstract: Gromov Witten theory counts complex curves in a Calabi Yau (CY) manifold. In many cases the manifold admits Landau Ginzburg phases, and the (LG) counting also enjoys symplectic and algebro-geometric constructions. Recently a new moduli space, called Mixed Spin P field, is provided to quantize the parameter linking CY to LG phases. The theory provides an algorithm recovering of Zinger's formula on g=1 quintic GW invariants. Another application is it proves finite generation conjecture of BCOV and Yamaguchi-Yau.

Date: 5/18, 10:00-11:30

Speaker: Hang Xue（University of Arizona）

Title: 退化情形的算术Gan-Gross-Prasad猜想

Abstract: 我们描述什么是算术Gan-Gross-Prasad猜想，然后在某个endoscopic的情形证明之.

Date: 5/11, 10:00-11:30

Speaker: Qingxue Wang（Fudan University）

Title: On the configuration space of flags and some applications

Abstract: This talk is an introduction to the configuration space of complete flags on a vector space and its Fock-Goncharov coordinates. We will explain how these coordinates are used to construct some invariants of 3-manifolds and elements in the 3rd Quillen K-group of the field of complex numbers.

Date: 4/27, 16:00-17:30

Speaker: Weiming Shen（Peking University）

Title: The Rigidity and Gap Theorem for Liouville's Equation

Abstract: In this talk, I will talk about the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying domain is a disc. More generally, we prove some gap theorems relating such a boundary integral to the number of components of the boundary. The conformal structure plays an essential role.

Date: 4/20, 10:00-11:30

Speaker: Jiangxue Fang（Capital Normal University）

Title: Introduction to GKZ-systems

Abstract: In this talk, I will study hypergeometric functions in the sense of Gel'fand, Kapranov, and Zelevinsky (now is called by GKZ-sytem). These functions generalize the classical hypergeometric functions of Gauss. In particular, I study the algebraic structure on the GKZ system by D-modules and perverse sheaves.

Date: 4/18, 10:00-11:30 （Different Date!）

Speaker: Yitao Wu（Yangzhou University）

Title: p-adic period maps

Abstract: In this talk, I'll give a brief introduction to the period maps and comparison theorems, which play a central role in the p-adic Hodge theory. I will also introduce some advanced topics base on the period maps.

Date: 4/13, 10:00-11:30

Speaker: Xun Yu（Tianjing University）

Title: Elliptic fibrations on K3 surfaces and Salem numbers of maximal degree

Abstract: We explain a characterization of the maximal Salem degree of automorphisms of K3 surfaces in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree 22. In particular, such automorphisms are not geometrically liftable to characteristic 0.

Date: 3/30, 16:00-17:30

Speaker: Raju Krishnamoorthy（Free Berlin University）

Title: An introduction to the companions conjecture

Abstract: We survey Deligne's "Companions conjecture" from Weil II, focusing on the crystalline side. We report on work using Abe's resolution of the companions conjecture over curves to build a correspondence between certain local systems and certain p-divisible groups on complete curves over F_q.

Date: 3/23, 16:00-17:30

Speaker: Xiaomeng Xu（MIT）

Title: Stokes phenomenon, quantum groups and 2d TFT

Abstract: This talk will include a general introduction to differential equations with singularities, and its relation with symplectic geometry, representation theory and 2d topologicial field theory. In particular, we will focus on the Stokes phenomenon of linear systems of ordinary differential equations, and construct the braiding of Drinfeld-Jimbo quantum groups as quantum Stokes matrices. We then study the isomonodromic deformation of Knizhnik–Zamolodchikov type equations, and relate its classical limit to the theory of Frobenius manifolds.

Date: 3/16, 16:00-17:30

Speaker: Xuezhang Chen（Nanjing University）

Title: Boundary Yamabe problem and related ones

Abstract: We start with the proof of a generalized Obata theorem on a compact manifold with boundary studied by Escobar (CPAM 90). This gives a natural motivation of the study of a Han-Li conjecture. Next we will mention our recent advance on this conjecture. Finally, we (joint with Mr. Yuping Ruan, an undergraduate student at nju) announce an improvement of Jin-Xiong's 2017 work, involving a conjecture of an isoperimetric inequality over scalar flat conformal class.

Date: 3/9, 16:00-17:30

Speaker: Shun Tang（Capital Normal University）

Title: Singular Lefschetz-Riemann-Roch theorem

Abstract: In this talk, I will introduce a Lefschetz-Riemann-Roch theorem for singular projective schemes which admit diagonalizable group scheme actions. I will also compare this result with a fixed point formula of Lefschtez type due to R. W. Thomason.