Algebraic Geometry of Integrable Systems and Singular Varieties

内文斯将进行两组项目相结合的想法，方法和问题的代数几何与表示论和数学物理。 第一个问题集将两种新的代数几何方法引入到可积系统的研究中，并提出解决这一领域的一系列既定问题;这些方法也有望阐明某些代数簇的几何，并为表示论提供一种新的工具。 第二个收集的问题集中在深化和推广的技术“motivic集成”从代数几何统一，扩展，并解释几个方面的几何奇异代数簇。多项式方程是最简单的类型的代数方程，但一套了解的解决方案，这样的方程shas显着的效用在各种各样的实际问题。 Nevins将应用多项式方程的几何研究技术，称为“代数几何”，以探索和解释数学物理中粒子系统和波动的行为。 在相反的方向上，他将扩展弦理论最近发展的方法，并将这些方法应用于阐明多项式方程解集的基本结构。

Nevins will carry out two clusters of projects combining ideas, methods,and problems of algebraic geometry with those of representation theory andmathematical physics. The first collection of problems introduces two newalgebro-geometric methods into the study of integrable systems andproposes to solve a collection of established problems in that area; thesemethods are also expected to illuminate the geometry of certain algebraicvarieties and provide a new tool in representation theory. The secondcollection of problems focuses on deepening and generalizing the techniqueof "motivic integration" from algebraic geometry to unify, extend, andexplain several facets of the geometry of singular algebraic varieties.Polynomial equations are among the simplest kinds of mathematicalequations, yet an understanding of the sets of solutions of such equationshas remarkable utility in a wide variety of practical problems. Nevinswill apply techniques from the geometric study of polynomial equations, afield known as ``algebraic geometry,'' to explore and explain the behaviorof both particle systems and wave motion in mathematical physics. In theopposite direction, he will expand on methods inspired by very recentdevelopments in string theory and will apply these methods to elucidatethe fundamental structure of the solution sets of polynomial equations.