Mathematical Sciences: Topology and Geometry of Algebraic Varieties

Pardon将研究代数簇(一些多项式方程的解集)的拓扑和几何性质，特别是当代数簇是奇异的(不是光滑的)时。在几何方面，他将研究这类空间上拉普拉斯算子的谱，以获得关于它们奇点的信息。特别感兴趣的是光谱不变量，它们被证明是有理的，因为它们最有可能是拓扑不变量。海恩计划在他的学生的帮助下，继续他对紧致黎曼曲面和紧致黎曼曲面上的点的构形进行参数化的空间拓扑的研究。该项目的主要目标是计算和/或获得关于这些模和配置空间的上同调的定性信息，包括平凡系数和扭曲系数。这些结果应该会在代数几何中产生几何结果，并可能通过共形场理论应用于粒子物理。主要使用的工具是霍奇理论和表象理论，并结合传统的几何方法。定义为多项式方程组的解集的空间的拓扑和几何是一个令人惊讶的丰富的学科，在数学和数学物理的许多不同领域都有应用。调查人员和他们的研究生们正在运用各种各样的技术。特别令人感兴趣的是建筑，这些建筑明确地依赖于空间的几何特征、面积、距离、曲率等，但将它们组合在一起的方式，可以证明只依赖于空间的拓扑结构，即空间的基本特征，以至于不会受到拉伸和扭曲其形状的影响，而不是切割或撕裂。

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.