Mathematical Sciences: Topology and Geometry of Algebraic Varieties

Pardon将研究代数变量（一些多项式方程的解的集合）的拓扑和几何性质，特别是当变量是奇异的（不光滑的）。在几何方面，他将研究这些空间上拉普拉斯算子的谱，以获得关于它们奇点的信息。特别有趣的是谱不变量，它被证明是有理的，因为它们最有可能是拓扑不变量。Hain，在他的学生的帮助下，计划继续他对空间拓扑的研究，这些空间的拓扑是参数化紧凑黎曼曲面和它们上点的构型。该项目的主要目标是计算和/或获得关于这些具有平凡和扭曲系数的模和组态空间的上同调的定性信息。这些结果将在代数几何中产生几何结果，并可能通过共形场论应用于粒子物理。所使用的主要工具是Hodge理论和表征理论结合传统的几何方法。将空间的拓扑和几何定义为多项式方程系统的解集是一门非常丰富的学科，在数学和数学物理的许多不同领域都有应用。研究人员和他们的研究生带来了惊人的各种各样的技术。特别有趣的是，明确依赖于空间几何特征的结构，面积，距离，曲率，等等，然而，以这样一种方式将它们组合起来，以证明只依赖于空间的拓扑，也就是说，空间的特征是如此基本，以至于不受拉伸和扭曲其形状的影响，缺乏切割或撕裂。

Pardon will study the topological and geometric properties of an algebraic variety (the set of solutions of some polynomial equations), particularly when the variety is singular (not smooth). On the geometric side, he will study the spectrum of the Laplace operator on such spaces in order to obtain information about their singularities. Of particular interest are spectral invariants which turn out to be rational, since they are most likely to be topological invariants. Hain, with the help of his students, plans to continue his study of the topology of spaces that parameterize compact riemann surfaces and configurations of points on them. The main objectives of the project are to compute and/or gain qualitative information about the cohomology of these moduli and configuration spaces, both with trivial and twisted coefficients. These results should yield geometric results in algebraic geometry, and may have applications to particle physics via conformal field theory. The main tools to be used are Hodge theory and representation theory combined with traditional geometric methods. The topology and geometry of spaces defined as solution sets of systems of polynomial equations is a surprisingly rich subject, having applications in many diverse areas of mathematics and mathematical physics. The investigators and their graduate students are bringing to bear a marvelously varied arsenal of techniques. Of particular interest are constructions that depend explicitly upon features of the geometry of a space, area, distance, curvature, and so forth, and yet combine them in such a way as to be provably dependent only on the topology of the space, that is on features of the space so basic as to be unaffected by stretching and distorting its shape, short of cutting or tearing.