Geometry and Topology of Singularities

提出的研究包括三个项目，集中在几何拓扑和代数几何的交界处的思想。第一个项目寻求复代数簇的Hodge理论不变量的刻画，重点是计算方面。潜在的应用包括格点计数问题的新的显式解和代数几何中的Novikov型猜想。在第二个项目中，PI提出了对复杂超曲面的某些整体解析不变量的详细研究，这些不变量衡量奇点的复杂性。文中还提出了与某些Calabi-Yau三叠体的Donaldson-Thomas理论的联系。在第三个项目中，PI旨在发展奇异复代数簇的等变特征类理论，并将其主要应用于奥布洛姆的特征类的计算。特别地，该部分的结果可用于计算奇异变种对称积的特征类的生成级数，推广和统一了文献中的许多已有结果。在不同但相关的脉络中，PI计划为定义在实数领域上的复变数发展合适的特征类理论；这种不变量编码复变数实点集上的深层次的拓扑和分析障碍，即具有实系数的多项式方程组的实解的存在性。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, focuses on ideas at the interface of geometric topology and algebraic geometry. The first project seeks characterizations of Hodge-theoretic invariants of complex algebraic varieties, with an emphasis on computational aspects. Potential applications include a new explicit solution to the lattice point counting problem, and a Novikov-type conjecture in algebraic geometry. In the second project, the PI proposes a detailed study of certain global analytic invariants of complex hypersurfaces which measure the complexity of singularities. A connection with the Donaldson-Thomas theory of certain Calabi-Yau trifolds is also suggested. In the third project, the PI aims to develop equivariant characteristic class theories for singular complex algebraic varieties, with primary applications to the computation of characteristic classes of orbifolds. In particular, the results sought in this part of the proposal can be used to compute generating series for characteristic classes of symmetric products of singular varieties, generalizing and unifying many of the existing results in the literature. In a different but related vein, the PI plans to develop suitable characteristic class theories for complex varieties which are defined over the field of real numbers; such invariants encode deep topological and analytical obstructions on the set of real points of a complex variety, i.e., on the existence of real solutions of a system of polynomial equations with real coefficients.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 evolved 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.