## 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 | Spring 2015

Date: 1/11, 3:00-4:30 (Different Date!)

Speaker: 刘东文（浙江大学）

Title: An introduction to the Parshin-Beilinson adeles and related topics

Abstract: In the 70s and 80s Parshin and Beilinson introduced the notion of higher dimensional adeles for quasicoherent sheaves, and proved that the cohomology of the associated adelic complex coincides with the sheaf cohomology. In this talk we give a brief survey of this theory as well as some recent developments, such as certain reciprocity laws on arithmetic surfaces.

Date: 12/25

Speaker: 许斌 (中国科学技术大学

Title: Reducible metrics and 1-forms on compact Riemann surfaces

Abstract: Does there there exist a conformal metric of positive constant curvature with finitely many conical singularities prescribed on a compact Riemann surface? This is a problem with a long history traced to H. Poincare and E. Picard. It is still widely open nowadays besides a lot of mathematicians obtained many classical results. Such a metric is called reducible if its developing map has diagonalized monodromy representation. The speaker will explain an observation in detail that all the information of a reducible metric can be encapsulated into some meromorphic 1-form with simple poles and with purely imaginary periods. Moreover, on the other hand, any such a 1-form gives naturally a 1-parameter family of reducible metrics, which form new examples of conical metrics of positive constant curvature. The speaker shall announce some new existence theorems about 1-forms with simple poles (and with purely imaginary periods) on compact Riemann surfaces and their two consequences as follows:

1. An explicit necessary and sufficient condition on the existence of reducible metrics on the Riemann sphere,

2. An explicity necessary and sufficient condition of prescribed conical angles under which there exists a reducible metric with those conical angles on some compact Riemann surface with genus g for each positive integer g.

Be sketched will the proof of the above existence theorem on the Riemann sphere in case that all the conical angles are pi times rationals. This special case also gives a new sufficient condition for the Hurwitz problem, which asks whether a admissible branch data is realized by some rational function. If time permitted, some problems interesting to the speaker will be proposed.

This is a joint work with Professor Qing Chen and my students Bo Li, Santai Qu and Ji-Jian Song. Since the beginning of this work, the speaker has been in a great debt to Professors Mao Sheng, Yifei Chen and Yingyi Wu at AMSS, to whom he express his deep gratitude.

Date: 12/22 （上午 10:00 - 11:30）

Speaker: 彭岳建 (湖南大学)

Title: Turan type of problems and Lagrangian method

Abstract: We will first review Turan type of problems. Then focus on the connection of Turan densities and Lagrangians of hypergraphs, and show some applications of this connection. We will also review some open problems.

Date: 12/4

Speaker: 许晨阳 （北京国际数学研究中心）

Title: Introduction to birational geometry

Abstract: Two varieties are birational equivalent if they have the same (field of) rational functions on it. This is one of the oldest topic in algebraic geometry. Although the study of surfaces was more or less done by Italian school about one century ago, the higher dimensional theory was only programmed about 30 years ago, called minimal model program (or Mori program). There has been lots of progress in this topic in recent years, but still many important questions remain open. I will try to outline some of them in the talk.

Date: 11/27

Speaker: 毛井（哈尔滨工业大学）

Title: Eigenvalue inequalities for the buckling problem of the drifting Laplacian on Ricci solitons

Abstract: In this talk, we focus on the buckling problem of the drifting Laplacian and give a general inequality for its eigenvalues on a bounded connected domain in complete Ricci solitons supporting a special function. By applying this general inequality, we obtain some universal inequalities for eigenvalues of the same problem on bounded connected domains in the Gaussian shrinking solitons and some general product solitons. This is a joint-work with Feng Du, Qiao-Ling Wang and Chuan-Xi Wu.

Date: 11/13, 16:00-17:30 (Different Time!)

Speaker: 张健(华南理工大学)

Title: Quantum determinants and quantum Pfaffians

Abstract: We use quantum exterior algebras to give a new and elementary formulation of quantum Pfaffians and generalized quantum Pfaffians based on quantum Plücker relations. In this approach, the quantum Pfaffians are for any square matrix satisfying a simple quadratic relation. In particular, we prove the fundamental identity expressing any quantum determinant as a quantum Pfaffian.

Date: 11/6

Speaker: Jilong Tong (University of Bordeaux, France)

Title: Fundamental groups of algebraic curves in positive characteristic

Abstract: Let $X$ be a proper connected smooth curve of genus at least two, defined over an algebraically closed field $K$. The algebraic fundamental group $\pi_1(X)$ of $X$ is a profinite group associated with $X$, which classifies the finite etale covers of $X$. When $K$ is of characteristic $0$ (for example when $K$ is the field of complex numbers), the structure of $\pi_1(X)$ is uniquely determined by the genus of $X$. On the other hand, if $K$ is of positive characteristic, the structure of $\pi_1(X)$ is much more involved. For example, A. Tamagawa proved that for an given profinite group $\Gamma$, there exist only finitely many isomorphism classes of smooth connected projective curves of genus at least two defined over the algebraic closure of a finite field, whose fundamental group is isomorphic to $\Gamma$. The aim of this talk is to explain some phenomenons in this direction.

Date: 10/30

Speaker: Jingzhou SUN (Stony Brook University)

Title: Introduction to ramdon complex geometry

Abstract: We will give a brief introduction to the subject of random complex geometry, including commonly-used techniques/tools, some interesting results and some unsolved problems. We will start with the very basics and then move slowly to the more advanced.

Date: 10/23

Speaker: Tatsuro Ito (安徽大学)

Title: Towards the classification of (P and Q)-polynomial association schemes

Abstract: This is a survey talk on the subject of the title above. In his lectures at Ohio State University in the late 70s, Eiichi Bannai proposed the classification of (P and Q)-polynomial association schemes; he regarded them as finite, combinatorial analogue of compact symmetric spaces of rank 1. I will trace the history back to the late 60s and explain how the concepts of P/Q-polynomial association schemes arose in relation to finite permutation groups, coding/design theory. I will then overview the progress of the classification in the 80s, 90s and thereafter. Finally I will present my personal view about the scope for the classification problem.

Date: 10/16

Speaker: 田可雷 (合肥工业大学)

Title: Introduction to KP hierarchy and its q-deformation

Abstract: Kadomtsev-Petviashvili hierarchy is an important integrable system. The talk aims to give a general introduction to KP hierarchy and its q-deformation. In particular, I will talk about Virasoro type algebraic structure hidden in the q-deformed KP hierarchy.

Date: 10/9

Speaker: 谈胜利 (华东师范大学)

Title: 代数曲面理论：历史与展望

Abstract: 代数曲线、代数曲面的发展历史，在数学史中占有非常重要的地位。我们将依照发展史，介绍数学中非常漂亮的代数曲面理论，并介绍它与其他分支的广泛联系，以及几个有趣的应用与未解决问题。

Date: 9/24, 2:00-3:30 (Note: Different Date and Time!)

Speaker: 葛根年(首都师范大学特聘教授、中国科技大学兼职教授)

Title: 组合数学与信息交叉科学简介

Abstract: 21 世纪是信息产业飞速发展的全新时代。随着传送数据的业务量及需求越来越大以及用户对双向多媒体通信的质量与速度的要求越来越高，通信传输技术正继续向高性能、低成本和智能化等主要方向发展，寻求新的信息传输算法与处理方式和物理实现是现代通信领域面临的一大挑战。与此同时，随着数据流量的增大和信息的争夺，通信容易被干扰和窃听，存在严重的安全隐患。在这样的大背景下，数学科学特别是离散数学有了更广阔的应用舞台。最新的研究表明，信号采样中的压缩感知，现代通信中的LDPC码以及多媒体传输中数字指纹认证对于组合数学的依赖是全方位的。组合数学的应用因此而显示出勃勃生机。报告人及其所带领团队多年来致力于利用深刻的代数、数论、概率论等数学工具，研究组合数学（组合设计、代数组合、极值组合等）、信息科学（编码密码学、信息安全、信息处理等）及其交叉科学中的前沿问题。报告人将通过一些具体事例来介绍组合数学与信息交叉科学的研究内容、存在问题及所用方法。

Date: 9/18

Speaker: Fucheng TAN (Shanghai Center for Mathematical Sciences and Shanghai Jiao Tong University)

Title: Crystalline comparison isomorphisms in p-adic Hodge theory

Abstract: In this talk, I will give a brief construction of the crystalline comparison isomorphisms for proper smooth formal schemes, and explain certain key techiniques. Such isomorphisms hold for coefficients and in the relative setting, i.e. for p-adic etale cohomology with coefficients in Z_p-local systems, and for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-etale site defined by Scholze. Another ingredient for the proof is the Poincare lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. This is part of a joint project with Jilong Tong.

Date: 9/11

Speaker: 林勇（中国人民大学）

Title: Homology and homotopy of digraphs

Abstract: We will give a new notion of homology and cohomology for digraphs and give some properties of these new graph homology theory. We will also show that the homology and cohomology for digraphs are homotopy invariant which is not true for classic clique homology on graphs.