Singular Spaces in Geometry and Topology

提出的研究包括三个项目，围绕几何拓扑学和代数几何交界处的思想展开。重点是了解奇点对复代数簇的几何和拓扑的影响。第一个项目详细研究了衡量奇点复杂性的局部完全交集的全局分析不变量(例如，特征数和类)。PI还提出了Steenbrink关于超曲面奇点的Hodge谱概念的特征类版本。第二个项目研究拟射影代数流形上点的Hilbert格式的拓扑和分析性质。这些模空间描述了给定空间上的点的集合(不一定是不同的)，它们揭示了所考虑的空间的几何和拓扑的看似隐藏的方面。这些模空间最初是在代数几何中研究的，它与数学的几个分支密切相关，如奇点、辛几何、表示理论甚至理论物理。PI的目的是得到这些(非常奇异的)模空间的特征类的生成级数公式。第三个项目研究了环簇的特征类，并将其应用于广义Pick-型公式和Euler-MacLaurin求和公式。环面簇作为复代数簇本身以及它们在凸多面体理论中的应用都很有趣。例如，计算凸多面体中格点的问题相当于计算某一环面簇的托德类。圆环簇的特征类公式经常转化为令人惊讶的数论恒等式(例如，用广义Dedekind和表示)，PI旨在详细研究这一恒等式。拓扑学是数学的一个分支，研究涉及位置和相对位置的几何图形的模式，而不考虑大小。从一开始，拓扑学就是在试图理解“奇异”(或不规则)空间的性质时产生的问题的影响下发展起来的。这种空间自然地出现在纯数学的各个领域，包括几何拓扑学、代数几何、数论以及更多的应用领域，如机器人运动规划的位形空间的研究。代数簇，即多项式方程的解的空间，是奇异空间的主要例子。它们是代数几何的主要研究对象，也为拓扑学理论提供了一个便利的试验场。提出的研究旨在提高我们对代数簇的拓扑性质的理解，这项任务通常涉及发现和研究各种不变量的局部和全局行为之间的微妙相互作用。

The proposed research, which includes three projects, is centered around ideas at the interface of geometric topology and algebraic geometry. The focus is on understanding the effect of singularities on the geometry and topology of complex algebraic varieties. The first project deals with a detailed study of global analytical invariants (e.g., characteristic numbers and classes) of local complete intersections which measure the complexity of singularities. The PI also proposes a characteristic class version of Steenbrink's notion of Hodge spectrum for hypersurface singularities. The second project studies topological and analytical properties of Hilbert schemes of points on a quasi-projective algebraic manifold. These are moduli spaces describing collections of (not necessarily distinct) points on a given space, which bring out seemingly hidden aspects of the geometry and topology of the space under consideration. These moduli spaces, originally studied in algebraic geometry, are closely related to several branches of mathematics, such as singularities, symplectic geometry, representation theory and even theoretical physics. The PI aims to obtain a generating series formula for characteristic classes of these (very singular) moduli spaces. The third project deals with a study of characteristic classes of toric varieties, with applications to generalized Pick-type formulae and Euler-MacLaurin summation formulae. Toric varieties are of interest both in their own right as complex algebraic varieties, and for their applications to the theory of convex polytopes. For instance, the problem of counting lattice points in a convex polytope amounts to the computation of Todd classes of a certain toric variety. Characteristic class formulae for toric varieties often translate into surprising number-theoretic identities (e.g., expressed in terms of generalized Dedekind sums), which the PI aims to investigate in detail.Topology is the branch of mathematics that studies patterns of geometric figures involving position and relative position without regard to size. From the very beginning, topology has developed under the influence of questions arising from the attempt to understand properties of "singular" (or irregular) spaces. Such spaces occur naturally in various fields of pure mathematics including geometric topology, algebraic geometry, number theory, and also in more applied fields, such as the study of configuration spaces for robot motion planning. Algebraic varieties, i.e., the spaces of solutions of polynomial equations, are major examples of singular spaces. They are the main objects of study in algebraic geometry, and also provide a convenient testing ground for topological theories. The proposed research aims to improve our understanding of topological properties of algebraic varieties, a task which often involves the discovery and study of subtle interactions between the local and global behavior of various invariants.