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` | 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` |

课程概况：

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.