Algebraic Geometry of Integrable Systems and Singular Varieties

内文斯将开展两组项目，将代数几何的思想、方法和问题与表示理论和数学物理的思想、方法和问题结合起来。第一个问题集将两种新的瓦尔格布罗几何方法引入到可积系统的研究中，并提出解决该领域的一系列既定问题；这些方法也有望阐明某些代数变体的几何，并为表示理论提供新的工具。第二部分的问题集中在深化和推广代数几何中的“动机积分”技术，以统一、扩展和解释奇异代数变量几何的几个方面。多项式方程是数学方程中最简单的一种，然而，对这类方程的解的集合的理解在各种各样的实际问题中具有显著的效用。内文斯将应用多项式方程的几何研究技术，即“代数几何”，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，他将扩展受弦理论最新发展启发的方法，并将应用这些方法来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.