Singular Spaces in Geometry and Topology

拟议的研究，其中包括三个项目，是围绕在几何拓扑和代数几何接口的想法。重点是理解奇点对复杂代数簇的几何和拓扑结构的影响。第一个项目涉及全局分析不变量的详细研究（例如，特征数和类）的局部完整的交叉，衡量的复杂性奇异。PI还提出了一个特征类版本的Steenbrink的概念霍奇谱的超曲面奇点。第二个项目研究拟投射代数流形上点的Hilbert格式的拓扑和分析性质。这些是模空间，描述给定空间上的（不一定是不同的）点的集合，这些点引出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，最初在代数几何中研究，与数学的几个分支密切相关，如奇点，辛几何，表示论，甚至理论物理。PI的目的是获得这些（非常奇异）模空间的特征类的生成级数公式。第三个项目是研究复曲面簇的特征类，并应用于广义Pick型公式和Euler-MacLaurin求和公式。复曲面簇是感兴趣的两个在自己的权利作为复杂的代数簇，并为他们的应用理论的凸多面体。例如，在凸多面体中计算格点的问题相当于计算某个环面簇的托德类。复曲面簇的特征类公式通常转化为令人惊讶的数论恒等式（例如，拓扑学是数学的分支，研究几何图形的模式，涉及位置和相对位置，而不考虑大小。从一开始，拓扑学就在试图理解“奇异”（或不规则）空间性质的问题的影响下发展起来。这样的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，例如机器人运动规划的配置空间的研究。代数变种，即，多项式方程的解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，也为拓扑理论提供了一个方便的试验场。拟议的研究旨在提高我们对代数簇拓扑性质的理解，这一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.