Mathematical Sciences: Topology and Geometry of Algebraic Varieties

赦免将研究的拓扑和几何性质， 代数簇（某些多项式的解的集合 方程），特别是当品种是奇异的（不光滑）。 在几何方面，他将研究拉普拉斯的谱 操作员在这样的空间，以获得有关其 奇点 特别感兴趣的是谱不变量 这是理性的，因为他们最有可能是 拓扑不变量 海恩在学生的帮助下， 继续他对空间拓扑的研究， 紧致黎曼曲面和其上的点的构形。 的 该项目的主要目标是计算和/或获得 关于这些模的上同调的定性信息， 配置空间，都平凡和扭曲的系数。 这些结果应该产生代数几何中的几何结果， 并且可以通过共形场应用于粒子物理学 理论 所使用的主要工具是霍奇理论和 表示理论与传统几何方法相结合。 定义为解集的空间的拓扑和几何 多项式方程组是一门非常丰富的学科， 在数学的许多不同领域都有应用， 数学物理 研究人员和他们的毕业生 学生们带来了各种各样的武器， 技术. 特别感兴趣的是依赖于 明确地基于空间，区域， 距离，曲率，等等，但联合收割机，他们在这样一个 以可证明地仅依赖于空间的拓扑的方式， 这是关于空间的基本特征， 拉伸和扭曲它的形状，没有切割或撕裂。

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.