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.
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 major goal of this research project is to enhance the understanding of topology of singular spaces and bridge gaps between different points of view coming from topology and algebraic geometry, respectively. The PI's work at the interface of several mathematical disciplines allows for fertile cross-pollination of tools and ideas. The outcomes of this project should be of equal interest for topologists and algebraic geometers: besides acquainting topologists with a variety of tools from singularity theory and algebraic geometry, the interdisciplinary nature of this project should lead algebraic geometers to become more acquainted with topological tools. The findings of this project will ultimately propagate to other fields where singularities play an important role, e.g., string theory and physics. During the period covered by this award (2010-2013), the PI accomplished the following: wrote 15 research articles (12 of which are published in prestigious mathematical journals, and 3 are in preprint form); organized 4 international conferences; edited 2 conference proceeding volumes; gave talks at 13 international conferences and 15 research seminars; mentored 3 graduate students. Overall, this project led to a better understanding of: the geometry and topology of singular hypersurfaces and complete intersections; topological and analytical properties of orbifolds (e.g., toric varieties, symmetric products and configuration spaces of varieties) and related moduli spaces (e.g., Hilbert schemes of points on algebraic manifolds); the development and study of new Poincare duality complexes. The project gave the PI the opportunity to travel and give talks at many international conferences and research seminars. New research collaborations have been established in USA, Germany, and Japan. The project also led to designing and teaching new graduate courses. Notes for these courses were written and made available to the public. More details about the Pi's research and teaching in connection with this award can be found at the url: www.math.wisc.edu/~maxim/
