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.
