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            1. Introduction to Mordern Mathematics

              August 12-14, 2011

              Venue: CMS 501

              Friday (August 12):
              09:15 - 10:15: Jianyi Shi
              10:30 - 11:30: Haibao Duan
              Lunch break
              14:00 - 15:00: Yuefei Wang
              15:15 - 16:15: Jianyi Shi
              Tea Break
              16:45 - 17:45: Fangyang Zheng

              Saturday (August 13):
              08:30 - 09:30: Nanhua Xi
              09:45 - 10:45: Xiaojun Huang
              11:00 - 12:00: Haibao Duan
              Lunch break
              13:30 - 14:30: Shengli Tan
              14:45: Tour to XiXi Wetland

              Sunday (August 14):
              08:30 - 09:30: Shengli Tan
              09:45 - 10:45: Yuefei Wang
              11:00 - 12:00: Shengli Tan
              Lunch break
              13:30 - 14:30: Xiaojun Huang

              Duan Haibao(段海豹),Academia Sinica
              Huang Xiaojun(黃孝軍),Rutgers University
              Shi Jianyi(時儉益),East Normal University
              Tan Shengli(談勝利),East Normal University
              Wang Yuefei(王躍飛),Academia Sinica
              Xi Nanhua(席南華),Academia Sinica
              Xiao Jie(肖杰),Tsinghua University
              Zheng Fangyang(鄭方陽),Ohio State University
              Zhu Xiping(朱熹平),Zhongshan University

              Scientific Organizers: Lizhen Ji(季理真) Kefeng Liu(劉克峰) Feng Luo(羅鋒) Shing-Tung Yau(丘成桐)

              Local Organizers: Fang Li(李方) Hongwei Xu(許洪偉)

              Local contact Persons: Qingyou Sun (qysun@cms.zju.edu.cn), Chunli Zhao (mathzhaocl@gmail.com)

              Description: This instructional conference consists of lecture series given by experts.It will cover some of the most important concepts, methods and theories in modern mathematics. Each lecture series will emphasize motivations of problems and results, connections with other subjects in order to present a historical perspective on the development of the topics under discussion.These seminar talks are especially designed for students and others who want to learn some new subjects from experts.

              Location: Center of Mathematical Sciences, Zhejiang University



              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.


              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.


              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


              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.


              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.


              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.


              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.

              Speaker: Huang Xiaojun,Rutgers University

              Title: Equivalence Problem in Complex Analysis and Geometry

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