【对外交流| 线上】2020年浙江大学求是数学暑期学校

浙江大学数学科学学院将于2020年7月4日-14日举办“2020年浙江大学求是数学暑期学校”。暑期学校共安排三门课程，每门课14--20课时，邀请了陈国璋教授（台湾清华大学，讲授Introduction to ergodic theory)、Alessandro Chiodo教授(法国索邦大学，讲授An invitation to algebraic geometry)、张希（中国科学技术大学，讲授An introduction to complex differential geometry）。

授课对象：数学专业大学本科二年级及以上求科班学生

主讲教师介绍：

  陈国璋 (Chen, Kuo-Chang)，台湾清华大学数学系教授，曾任台湾清华大学数学系主任。研究领域为动力系统、天体力学及微分方程。利用微分方程的方法在多体问题上获得重要突破，研究成果发表于Annals of Math.，Comm. Math. Phys.，Amer. J. Math.，Arch. Ration. Mech. Anal. 等国际顶尖期刊，获得多项学术荣誉及奖励。现担任学术期刊Nonlinearity及Discrete and Continuous Dynamical Systems-Series A编委。

Alessandro Chiodo教授, 法国索邦大学(巴黎第六大学)教授，国际著名代数几何专家，研究成果发表于Invent. Math.，Adv. Math.， J. Eur. Math. Soc.等国际顶尖期刊。

张希，中国科学技术大学教授、博士生导师、国家杰出青年基金获得者、中国科学院“百人计划”入选者。主要从事非线性微分方程及其在微分几何、复几何中的应用等方面的研究。 例如: (1) 全纯丛上典则度量的存在性及相关热流方程方面的研究; (2) 复Monge-Ampere方程及其应用。研究与数学的许多分支相关，如：偏微分方程，代数几何，规范场论，微分几何，复几何；是基础数学领域中的交叉学科。其研究特点是运用多种非线性分析方法与代数几何、多复变中的经典方法相结合来研究微分几何和复几何中的一些重要问题。

课程安排：

课程1：An introduction to complex differential geometry，

任课老师：张希教授

钉钉群

助教：夏树灿

课程2： An invitation to algebraic geometry，  

任课老师：Alessandro Chiodo教授，

Zoom Meeting ID：7597950577，https://zoom.com.cn/j/7597950577?pwd=akYxditiWDcreDBkM1BsUzlXdFliQT09

助教： 金理泽

课程3：Introduction to ergodic theory，

任课老师：陈国璋教授

腾讯会议：https://meeting.tencent.com/p/7567591902，会议 ID：756 759 1902，会议密码：347088   

  助教： 吴家彦

课程介绍：

An introduction to complex differential geometry

任课老师(instructor): 张希(Zhang Xi)

1, 课程大纲(Syllabus)

   There has been fundamental progress in complex differential geometry in the last three decades, non-linear analysis method have been found very useful in the study of complex differential geometry.  The aim of this course is to give an essentially self-contained introduction to the basic theories, concepts of complex differential geometry, and present the students with some advanced topics in Kaehler geometry specially in the theory of canonical metrics and nonlinear PDE on complex manifolds. This course include three topics. The first topic include the basic theories, concepts of complex geometry and Kaehler geometry, for example: almost complex structure, complex manifold, Chern class, holomorphic vector bundle, Hermitian and Kaehleian structure, the curvature of Kaehler metric, Hodge theory. The second topic will focus on  Calabi-Yau theorem and complex Monge-Ampere equation. In the lastly topic, we introduce the  Hermitian-Einstein equation and its applications.

This course shoud be suitable for senior undergraduates and graduated students with a basic knowledge in complex anlaysis, PDE and differential manifold.

2, 章节(Chapter/Session)

   1,Basic concepts in complex geometry(8 hours)

(1), holomorphic map

(2), complex manifold

(3), holomorphic vector bundle

(4), Hermitian metrics and Kahler structures

(5), kahler manifold

(6), Vanishing theorems

2, Calabi-Yau theorem （6 hours）

   (1), Complex Monge-Ampere equation

   (2), Calabi conjecture

   (3), C^0-estimate

   (4), C^2-estimate

3, Hermitian-Einstein metrics on holomorphicvector bundles （6 hours）

   (1), Stability by Mumford

   (2), Hermitian-Einstein connection

   (3), Donaldson’s functional

   (4), Donaldson’s heat flow and Yang-Mills flow

   (5), Donaldson-Uhlenbeck-Yau theorem

3,参考书目(References)

1, Lectures on Hermitian-Einstein metrics for stable bundles and Kahler-Einstein metrics, by Yum-Tong Siu;

2,  Differential geometry of complex vector bundles, by S.Kobayashi;

3, Complex geometry, D. Huybrechts;

4, Complex analytic and differential geometry, by, J.P.Demailly

5, Canonical metrics in Kaehler geometry. By Gang.Tian

6, Differential analysis on complex manifolds, by O.Wells

7, Principles of algebraic geometry, by Griffiths andHarris.

8, Complex manifolds, J.Morrow and K.Kodaira,

9，Hodge theory and complex algebraic geometry, C.Voisin.

An invitation to algebraic geometry.

任课老师(instructor): Alessandro Chiodo

Schedule. 4,6,7,8,9,13,14/7/2020 (seven 1h30 lectures starting from 3pm to 4:30 pm Hangzhou time)

Lecture 1 - 4/7 - The genus of a curve.

Lecture 3 - 6/7 - Riemann-Roch

Lecture 4 - 7/7 - Serre duality

Lecture 5 - 8/7 - The canonical map

Lecture 6 - 9/7 - Riemann-Hurwitz

Lecture 7 - 13/7 - Abel theorem

Lecture 7 - 14/7 - Final comments

Abstract: The course introduces some of the main theorems of algebraic geometry in the case of smooth complex curves. The plan is to gradually move from the simplest objects to the study of maps between them, and prepare the students to a future systematic study of singular and higher dimensional objects. The lectures may be also attended to simply get a gist of the subject. We provide simple proofs of these theorems based on the knowledge of an undergraduate student in his third year: we essentially rely on some knowledge of complex differential forms, which we will systematically recall and recast in our context.

Introduction to Ergodic Theory

任课老师(instructor): 陈国璋 (Chen, Kuo-Chang)

Abstract:

Ergodic theory is the mathematical subject which studies long-term

average behavior of measurable dynamical systems. It has deep

connections with statistical physics, functional analysis, number

theory, probability, differential geometry, among others. This short

course is a brief introduction intended for students who are interested

in dynamical systems and analysis. Audiences are assumed to have some

basic knowledge in real analysis.

Schedule:

There will be 5 lectures, 3 hours for each lecture:

Lecture 1. Invariant and ergodic measures

Lecture 2. Ergodic theorems

Lecture 3. Mixing properties

Lecture 4. Recurrence

Lecture 5. Entropy

Primary References:

1. K.E. Peterson, Ergodic Theory, Cambridge University Press, 1983.

2. M. Pollicott and M. Yuri: Dynamical Systems and Ergodic Theory,

London Mathematical Society, Cambridge University Press, 1998.

3. P. Walters: An Introduction to Ergodic Theory, Graduate Texts in

Mathematics, Springer-Verlag, 1982.