# 研究生教育创新计划高水平学术前沿讲座【Andriy Haydys】

 `时 间` `9月26日``15:00-16:30` `9月28日``15:00-16:30` `9月29日``15:00-16:30` `9月30日``15:00-16:30` `10月8日``15:00-16:30` `10月9日``15:00-16:30` `教 室` `5407` `5107` `5507` `5407` `2209` `2209`

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.