Geometry and Topology of Singularities

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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.