Integrable systems and special Kahler geometry

教师介绍
本讲教师：Andriy Haydys 课程介绍
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 timevariable 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 2ndimensional manifold M equipped with a nondegenerate (in a certain sense) 2form ω, which is called a symplectic form. An integrable system can be described as a fibration π : M →B over an ndimensional base B such that, roughly speaking, for any point b?B the fiber Mb := π1(b) is Lagrangian, i.e., ?*bω= 0, where ?b: 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 hyperK?hler) 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 K?hler structure. The main point for the last part of the lectures will be to explain a relation between complex integrable systems and special K?hler geometry and to outline some questions for further research.

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