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Speaker: Jian-yi Shi (East China Normal University)

Title: Reflection groups and some related topics.

Abstract： I shall give a brief introduction for the theory of reflection groups. This includes Coxeter groups, complex reflection groups, Iwahori-Hecke algebras and cyclotomic Hecke algebras, etc,and their application to various areas of mathematics.

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Speaker:Haibao Duan (Academia Sinica)

Title: Schubert calculus, I, II.

Abstract: Clarifying Schubert's enumerative calculus was the contents of Hilbert's 15th problem, and has also been a major theme of twentieth century algebraic geometry.

We recall the historic developement of the subject, emphasize its algorithmic aspects that are of current interest, and explain its relevance with cohomology theory of Lie groups.

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Speaker:Sheng-Li Tan (East China Normal University)

Title:Mathematics from lower degrees to higher degrees

Abstract:Degree is an important concept accompanied with the development of mathematics. In this short course, we provide an introduction to Algebraic Geometry and some related fields in modern mathematics, e.g., geometry, theory of functions and number theory. We will follow the history of mathematics, starting from lower degrees and dimensions to the higher ones. We are going to focus on its origin and development，the classical and modern methods, and the main problems and their progresses.

Part One: Mathematics of degree one and two

1） Pappus Theorem, Pascal Theorem and their applications in plane geometry

2） Complex numbers, points at infinity and the homogeneous coordinates

Part Two: Mathematics of degree three

3） Chasles Theorem on cubics (a generalization of Pascal Theorem), Bezout Theorem and Noether’s Fundamental Theorem

4） Group structure on cubics and Poncelet’s Theorem on conics

5） Computing rational points on conics and cubics

6） Rational right triangles, congruent numbers and BSD conjecture

7） Graphs of complex algebraic curves, genus formula and the non-integrability of elliptic integrals

8） Trigonometric functions, elliptic functions and algebraic functions (Riemann’s theory)

9） Riemann-Roch Theorem and the classification of algebraic curves

Part Three: Mathematics of higher degrees

10） Generalize problems to higher degrees and higher dimensions

11） Algebraic functions of two variables and the Riemann-Roch problem

12） Graphs of algebraic surfaces and Poincare’s algebraic topology

13） Chern numbers and Miyaoka-Yau inequality

14） Classification of algebraic surfaces, K3 surfaces and Calabi-Yau manifolds

15） Method of abstract algebra (geometrization of algebra)

16） Singularities and their resolution (application of the method of algebra)

17） Diophantine geometry and Arakelov geometry (geometrization of number theory)

18） On some open problems on polynomials

19） Applications and further development of algebraic geometry

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Speaker:Xi Nanhua(Academia Sinica)

Title: Lusztig's a-function for Coxeter groups

Abstract: Lusztig's a-function for Coxeter groups plays an important role in Kazhdan-Lusztig theory. In this talk we will give some discussion to the function. The contents include definition of the function, basic properties, some recent progresses.

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Speaker:Zhu Xiping(Zhongshan University)

Title:(1)The Ricci Flow and Its Applications (I)

(2)The Ricci Flow and Its Applications (II)

Abstract:In these two talks we will consider the Ricci flow and its geometric applications. We will discuss the short-time existence, uniqueness, curvature estimates, singularities and long-time behaviors. We will also discuss some applications in the classifications of positively curved Riemannian manifolds.

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Speaker:Zheng Fangyang,Ohio State University

Title: On nonnegatively curved Kahler manifolds.

Abstract: In this talk, we will recall the developement on the topic of nonnegatively curved Kahler manifolds in the past 30 years, starting from the pioneer work of Mori and Siu-Yau on Frankel/Hartshorne Conjectures. This topic can be regarded as the elliptic case of the uniformization theory, which attempts to generalize the classic Riemann Mapping Theorem in higher dimensions. We will survey some of the major developments in this area, and also discuss some open questions towards the end, including the so-called Generalized Hartshorne Conjecture.

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Speaker: Wang Yuefei,Academia Sinica

Tittle. Introduction to complex dynamics

Abstract. The dynamics of holomorphic maps on complex manifolds occupies a distinguished position in the general thoery of smooth dynamcal systems. A broad spectrum of theories, such as quasi-conformal mappings, Teichmuller spaces, hyperbolic geometry, potential theory, Kleinian groups and algebraic geometry, etc are closely related and interacted. We will give an introduction to this topic.

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Speaker: Huang Xiaojun,Rutgers University

Title: Equivalence Problem in Complex Analysis and Geometry

Abstract:

We present an elementary discussion for the equivalence problem in complex analysis and geometry, both through the approach of algebraic and geometric method.