Mathematical Sciences: "Algebraic Geometry of Nonlinear Integrable Systems"

主要研究完全可积系统及其超对称扩展理论中的问题，以及由积分系统引起的常微分方程理论中的问题。研究的目的是确定可积系统无穷维解集的代数结构，并将该结构与代数曲线的模空间联系起来。一个完全可积系统是空间中每个点的几何和代数数据的集合，它定义了一个曲面的集合，这些曲面填满了空间，就像书中的书页填满了空间一样。一般来说，书页或书页不像书上那样是平的，而是弯曲的，就像人们把书扭曲时那样。叶片的扭曲方式由可积系统提供的数据来描述。系统的曲面或叶可以认为是微分方程的解集，不同的叶对应于不同的初始条件。首席研究员将尝试用代数几何的方法来理解解空间。

The principal investigator will study problems in the theory of completely integrable systems and their supersymmetric extensions as well as problems in the theory of ordinary differential equations arising from the integral systems. The goal of the research is to determine the algebraic structure of the infinite dimensional set of solutions to an integrable system and relate this structure to the moduli space of algebraic curves. A completely integrable system is a collection of geometric and algebraic data at each point in a space which defines a collection of surfaces which fill up the space much as the pages in a book fill space. In general the pages or leaves are not flat as they ordinarily would be in a book but curved as they would be if one twisted the book. The way in which the leaves are twisted is described by the data provided by the integrable system. The surfaces or leaves of the system may be thought of as the solution set of a differential equation with different leaves corresponding to different initial conditions. The principal investigator will attempt to understand the solution spaces by using methods of algebraic geometry.