## 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 | Spring 2014 | Fall 2014 | Spring 2015 | Fall 2015 | Spring 2016 | Fall 2016

Date: 6/23, 16:00-17:30

Speaker: Ljudmila Kamenova (Stony Brook University)

Title: Hyperbolicity in hyperkaehler geometry

Abstract: The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds, and in particular,Calabi-Yau manifolds are Kobayashi non-hyperbolic. Using ergodicity of complex structures, together with S. Lu and M. Verbitsky we prove this conjecture for all K3 surfaces and for most classes of hyperkaehler manifolds. In the talk I will also give the algebraic version of hyperbolicity. Together with M. Verbitsky we prove that projective hyperkaehler manifolds with Picard rank at least two are algebraically non-hyperbolic.

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

Speaker: Jason Starr (Stony Brook University)

Title: The legacy of Zeng Jiongzhi

Abstract: Zeng Jiongzhi, known in the west as Chiungtze C. Tsen, proved spectacular results during his tragically short career. I will recount Tsen's famous theorems about existence of solutions of polynomial equations, as well as the corollaries of these theorems to Brauer groups and other questions in algebra. Then I will discuss the legacy of Tsen's work: rationally connected varieties, Serre's conjectures about principal bundles for semisimple groups, and rational points.

Date: 6/9, 16:00-17:30

Speaker: 訚琪峥（北京国际数学中心）

Title: Curves，sheaves，and cycles on K3 surfaces

Abstract: We study algebraic cycles on K3 surfaces via curves and sheaves. We review Beauville and Voisin's work on the Chow ring of K3 surfaces, and discuss some recent progress in joint work with Rahul Pandharipande, Junliang Shen, and Xiaolei Zhao.

Date: 5/19, 16:00-17:30

Speaker: 徐浩（浙江大学）

Title: Heat kernels coefficients and spectral geometry

Abstract: Short-time asymptotic expansion of the heat kernel played an important role in spectral geometry. We survey methods of computing the asymptotic expansion coefficients of heat kernels and their applications in spectral geometry. We will present our work on Feynman diagram formulas for the coefficients in the asymptotic expansion of heat kernels on Kahler manifolds.

Date: 5/12, 16:00-17:30

Speaker: 刘佳伟（北京大学）

Title: K\"ahler-Ricci flow: from twisted case to cusp.

Abstract: In this talk, I will show the long-time existence, uniqueness, regularity and convergence of conical K\"ahler-Ricci flow with weak initial data which admits L^p-density for some p>1. Furthermore, I will talk about cusp K\"ahler-Ricci flow which is the limit flow of conical K\"ahler-Ricci flows when the cone angles tend to zero.

Date: 5/5, 16:00-17:30

Speaker: Ariyan Javanpeykar（University of Mainz, Germany）

Title: Finiteness results in number theory

Abstract: The set of rational points on an elliptic curve over a number field is finitely generated by a theorem of Mordell and Weil. In these lectures, we will first explain the main ideas of the proof of the latter theorem. A part of the proof of the Mordell-Weil theorem fits in well with a philosophy of Shafarevich. Namely, the set of "objects of fixed type" over a fixed number ring should be finite. I will give precise examples illustrating Shafarevich's philosophy with abelian varieties, hyperbolic curves, and K3 surfaces and explain the relation to the Lang-Vojta conjecture. In the course of lectures, the proof of Faltings' finiteness theorem will be also mentioned.

Date: 4/28, 16:00-17:30

Speaker: 张会春（中山大学）

Title: Weyl's asymptotics for eigenvalues on non-smooth setting

Abstract: One of the most fundamental theorems in spectral geometry is the Weyl's law: on any closed n-dimensional Riemannian manifold, we have a leading asymptotic for eigenvalues of the Laplace operator. In this talk, we will introduce an extension of the Weyl's law to metric measure spaces with generalized Ricci curvature bounded from below. This is a joint work with Prof. Xi-Ping Zhu.

Date: 4/21, 10:00-11:30 @ 1418（Different Time and Room!）

Speaker: 王善文（复旦大学）

Title: p-adic modular forms

Abstract: In this talk, we will give a short introduction to the theory of p-adic modular forms and discuss some arithmetic applications.

Date: 4/14, 16:00-17:30

Speaker: 朱文光（中国科学技术大学物理系）

Title: Theoretical Design of 2D Ferroelectric and Multiferroic Materials

Abstract: In this talk, we will first present a discovery of a 2D ferroelectric materials family, rooted in III2-VI3 van der Waals materials and distinctly characterized by exhibiting versatile ferroelectricity with out-of-plane electric polarization. The device potentials of these new 2D ferroelectric materials will be demonstrated using the van der Waals heterostructures. Based on these discovered 2D ferroelectric materials, we further propose a design of 2D multiferroic materials via doping a small amount of magnetic ions into a 2D ferroelectric material. This design provides a novel multiferroic system based on van der Waals 2D materials, and these 2D multiferroic materials are expected to have strong coupling between the ferroelectric polarization and the magnetic order.

Date: 4/7, 14:00-15:00（Different Time!）

Speaker: 华波波（复旦大学）

Title: 平面铺砖问题的推广

Abstract: 我们研究推广的平面铺砖问题：用边长为1的正多边形作为砖块，将平面(在拓扑意义下)铺满，使得每个顶点处的顶角和不超过360度。这样的铺砖方式的组合和几何限制是什么？如何实现？

Date: 3/31, 16:00-17:30

Speaker: 麻希南（中国科学技术大学）

Title: 单相自由边界问题的由来

Abstract: 我们讨论单相自由边界问题的由来，研究方法以及最近的几个进展，我们的演讲主要参考Alt-Caffarelli（crelles，81年）的论文。

Date: 3/24, 16:00-17:30

Speaker: 陈小杨（同济大学）

Title: An Introduction to Polar Actions

Abstract: An isometric group action on a Riemannian manifold $(M, g)$ by a compact Lie group $G$ is called polar action if there is a connected, closed, embedded submanifold of $M$ which meets all the orbits orthogonally. There is a rich theory of planar actions. We will give an introduction to this rich theory.

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

Speaker: 吴鹏（上海数学中心）

Title: Rigidity of Einstein four-manifolds with positive sectional curvature

Abstract: Einstein metrics are most natural Riemannian metrics on differentiable manifolds. In dimensions 2 and 3, they must have constant sectional curvature, while in dimension 4, they are much more complicated. For the complex setting, in 1990 Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, and in 2012 LeBrun classified Hermitian, Einstein four-manifolds with positive scalar curvature. For the real setting, however less is known, even assuming a (strong) condition of positive sectional curvature. In this talk I will first talk about some background on Einstein manifolds, then I will focus on Einstein four-manifolds with positive curvature. If time permitted, I will also talk about my recent attempts of attacking this problem via k-positive curvature operator.

Date: 3/13(Monday), 9:00-10:30(Different Time!)

Speaker: 张磊（陕西师范大学）

Title: Some problems related to abundance for for 3-fold in char p

Abstract: Recently, Birkar, Hacon and Xu have proved existence of minimal model of 3-folds in char p >5.
We focus on abundance 3-folds in char p >5: for a minimal klt 3-fold X, is the canonical divisor K_X semi-ample?
According to whether X has non-trivial Albanese maps or not, we fall into two cases.
Some progresses have been made for those varieties with non-trival Albanese maps in the recent years. In this case to prove abundance it suffices to prove
(1) characterization of abelian varieties by the condition Kodaira dimension =0, that is, if a variety X has maximal Albanese dimension and Kodaira dimension zero, is X birational to an abelian variety.
(2) Iitaka conjecture: for a fibration f: X ---> Y of smooth varieties, it is conjectured thatk(X) >= k(Y) + k(F) where F denotes the generic fiber.
We will discuss the two problems.
Very little is known for varieities with trival Albanese maps, if time permitting we will discuss this case and explain the main obstacles in this case.

Date: 3/10, 16:00-17:30

Speaker: 余君（北京国际数学中心）

Title: Antipodal sets in symmetric spaces

Abstract: An antipodal set in a symmetric space is a set such that all points are fixed points of the geodesic symmetry with center any point of this set. We discuss a way of classifying antipodal sets. For irreducible symmetric spaces, we give the precise classification.

Date: 2/24, 16:00-17:30

Speaker: 刘文飞（厦门大学）

Title: Effective Boundedness of Stable Surfaces

Abstract: Stable surfaces, as the two-dimensional analogue of stable curves, were introduced by Kollár and Shepherd-Barron in 1988 to compactify the moduli spaces of surfaces of general type in a geometrically meaningful way. Nowadays the theory of stable surfaces, including the existence of various bounds as an important aspect, has been well established. In this talk I will give an introduction to the notion of stable surfaces and present some effective bounds concerning their geography and pluricanonical maps, based partly on joint works with Sönke Rollenske and with Valery Alexeev.

Date: 2/17, 16:00-17:30

Speaker: 冯仁杰（北京国际数学中心）

Title: Random fields and geometry

Abstract: I this lecture, I will talk about some classical topics and methods in random geometry. One of the most important topics is the random topology of the random fields such as the random Euler characteristics which has deep applications in astrophysics such as CMB problem. Many methods such as the Kac-Rice formula, tube formula, Gaussian kinematic formula are developed. I will also talk about some most recent results and some open questions.