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研究生教育创新计划高水平学术前沿讲座【Andriy Haydys】

课程名称:Integrable systems and special Kähler geometry
授课教师:Andriy Haydys  (University of Freiburg)
授课地点:第五教学楼
授课时间:
 
时 间
926
15:00-16:30
928
15:00-16:30
929
15:00-16:30
930
15:00-16:30
108
15:00-16:30
109
15:00-16:30
教 室
5407
5107
5507
5407
2209
2209

课程简介:

Roughly speaking, an integrable system is a system of ordinary differential equations, which can be integrated by means of first integrals, i.e., functions remaining constant in the time-variable along any solution. I will, however, emphasize a more geometric approach to the problem of integrating ODEs stemming from the classical Hamiltonian mechanics. In this approach one is interested in a 2n-dimensional manifold M equipped with a nondegenerate (in a certain sense) 2-form ω, which is called a symplectic form. An integrable system can be described as a fibration π : M →B over an n-dimensional base B such that, roughly speaking, for any point bϵB the fiber Mb := π-1(b) is Lagrangian, i.e., ɩ*bω= 0, where ɩb: Mb→M is the canonical embedding. 
The main point of the first part of the lectures (roughly 3 lectures) is to explain this geometric approach in some details. While the material of the first part is well known, this will serve us as a model for what will come in the second part and is much less well understood. 
In the second part I will describe a complex version of integrable systems, which is essentially a complex symplectic (also known as hyperKähler) manifold equipped with the structure of a holomorphic Lagrangian fibration. The situation becomes much more rigid in this case and it turns out that the base of a holomorphic Lagrangian fibration can be equipped with a so called special Kähler structure. The main point for the last part of the lectures will be to explain a relation between complex integrable systems and special Kähler geometry and to outline some questions for further research.  

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中科院数学与系统科学研究院 | 北京国际数学研究中心 | 安徽省数学会