## 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) etc.

## Upcoming Talks

## Past Talks

Archive of GAP Seminar: Fall 2013 | Spring 2014 | Fall 2014

Date: June 1, 2015

Speaker: Renjie FENG（北京国际数学中心）

Title: Extrema of random holomorphic fields

Abstract: One of the main motivation in random geometry is to study the generic geometry properties. In the first part of the talk, I will define the random holomorphic fields and exhibit several well-known results regarding generic geometry behaviors. In the second part, two of my results on the extrema of random fields will be given: the distribution of critical values and the mean value of the supremum of L^2-normalized fields. I will also discuss several open problems.

Date: May 25, 2015

Speaker: Yan WANG（Georgia Institute of Technology）

Title: Induced forests in bipartite planar graphs

Abstract: The Four-Color Theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary do not share the same color. The Four-Color Theorem was proved by using discharging method with the aid of computer. Nowadays, discharging method is a standard technique to tackle planar graph problems. In this talk, we will introduce the discharging method and then apply it to show that every simple bipartite planar graph on n vertices contains an induced forest on at least (4n+3)/7 vertices. This improves the best-known lower bound for Akiyama and Watanabe's conjecture (Every simple planar bipartite graph on n vertices contains an induced forest on at least 5n/8 vertices).

Date: May 18, 2015

Speaker: 薛江维（武汉大学）

Title: CM-fields as endomorphism algebras of abelian varieties over finite fields

Abstract: In this talk, we will review the Honda-Tate theory and try to answer the following question: when is a CM-field the endomorphism algebra of an abelian variety definite over a finite field.

Date: May 11, 2015

Speaker: 马瑶（东北师范大学）

Title: The cohomology and 1-parameter formal deformations of Hom-Lie triple systems

Abstract: In this talk, I will introduce the cohomology theory, by taking Lie algebras as an example. After that I will focus on the cohomology and 1-parameter formal deformations of Hom-Lie triple systems, which is a joint work with Liangyun Chen and Jie Lin.

Date: April 27, 2015

Speaker: 来米加（上海交通大学）

Title: On integral curvature pinching 3-sphere Theorem

Abstract: In this talk, I will first review several celebrated sphere theorems, then discuss my work on an integral curvature pinching 3-sphere theorem. For the proof, we shall focus on a technique from conformal geometry.

Date: April 20, 2015

Speaker: 葛健（北京国际数学中心）

Title: 1/4-pinched contact sphere theorem

Abstract: In this talk I will present a proof of 1/4-pinched sphere theorem, which says if the curvature of a contact 3-manifold with compatible Riemannian metric is 1/4-pinched, then the contact structure is universal tight. We also have some results for open manifolds. This is a joint work with Yang Huang.

Date: April 13, 2015

Speaker: 赵立璐（合肥工业大学）

Title: The Gauss circle problem and related topics

Abstract: Let $\mathcal{N}(R)$ denote the number of integral points in the circle $$\{(x,y)\in \R^2: x^2+y^2\le R^2\}.$$ The classical result of Gauss asserts that $$\mathcal{N}(R)=\pi R^2+ E(R),$$ where the error term $E(R)$ satisfies $|E(R)|\le 2\sqrt{2}\pi R$. The Gauss circle problem is to investigate the asymptotic behavior of $E(R)$. We shall give a survey of developments on the research of the Gauss circle problem and related topics such as the Dirichlet divisor problem and the theory of Riemann zeta function.

Date: April 7 (Different Date!) 2015

Speaker: 陈洪佳（中国科学技术大学）

Title: Introduction to Lie algebras and their representations

Abstract: Lie algebra is an important basic subject in mathematics, a sound knowledge of it being a must for research in many diverse areas. In this talk, I will give a general introduction to Lie algebras, based on the smallest simple Lie algebra sl2. In particular, I will talk about simple (non)weight modules, universal enveloping algebra, PBW theorem, the Casimir element and try to give a description of all simple modules over sl2. I will end this talk by focusing on a recent work about representations over sl2 and some related Lie algebras, which is a joint work with one of my student.

Date: March 23, 2015

Speaker: 宁博（天津大学）

Title: Some Topics on Spectral Extremal Graph Theory

Abstract: Spectral extremal graph theory is a beautiful branch of graph theory which have attracted much attentions recent decades. In particular,
the following Tur\'{a}n-Brudali-Solheid-type problems are considered in this talk: for a given graph $H$, what is $\max\{\rho(G): G~\mbox contains~no~H,v(G)=n\}$? We will briefly survey some interesting and basic theorems in this area, such as spectral analogues of Mantel's theorem on triangle, Tur\'{a}n's theorem on clique, Ore's theorem on Hamilton cycle, Bollob\'{a}s' theorem on pancyclicity, Reidei's theorem on quadrangle, Erd\"{o}s' conjecture on even cycle and etc. Besides, we will also compare these results with the corresponding Tur\'{a}n-type theorems in extremal graph theory. In particular, some of our recent works on spectral analogues of classical theorems on Hamilon cycles (Erd\"{o}s (1962), Moon and Moser (1963), Matthews and Sumner (1985)) will be presented. The main tools involve kinds of closure theories, Tur\'{a}n's theorem, solutions to Zarankiewicz problem and spectral inequalities. (Some results are based on recent joint works with Dr. Jun Ge (LMA, 2015) and Dr. Binlong Li (Preprints, 2015))

Date: March 16, 2015

Speaker: 徐国义（清华数学中心）

Title: Three Circles Theorems for harmonic functions

Abstract: I will present my recent work on harmonic functions over manifolds with non-negative Ricci curvature. I will try to tell the main line of the story about the harmonic functions on manifolds since Yau's work, and explain the intuitive idea behind the proof of our main results （including the existence of nontrivial harmonic functions of polynomial growth, the uniform bound of the frequency of those harmonic functions）. And suitable technical stuff will presented, but elementary differential geometry and PDE knowledge is enough for understanding most part of the talk.

Date: March 9, 2015

Speaker: 张毅（复旦大学）

Title: Hodge metric and Hyperbolicity

Abstract: We will discuss the construction of a Kaehler metric on a smooth quasi-projective manifold $S$ which effectively parameterizes polarized projective manifolds with semi-ample canonical line bundles, and its application to hyperbolicity.