The Topology and Hodge Theory of Algebraic Maps

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The award supports research in the field of algebraic geometry, the discipline devoted to the study of polynomial or algebraic equations. Algebraic equations are both beautiful and ubiquitous, as they describe many natural phenomena, from the motion of planets to the shape of leaves and flowers, to the behavior of microscopic particles. The goal of this research project is to study the deeper properties of the solutions to more complicated algebraic equations (called algebraic maps). The investigator plans to continue long-term investigation of the topology, Hodge theory, and cycle theory of algebraic maps. The close connection between the two main threads of the research, namely the discovery of new and deep aspects of the general theory and the study of fundamental examples, are the motivating principle behind the work. It is anticipated that the results will be of use to mathematicians (in algebraic geometry, combinatorics, and representation theory) and to mathematical physicists (in string theory). The investigator will explore the fundamental aspects of the general theory as well as important examples through five projects: to determine the supports of Hitchin fibrations in the case of the general linear group; to compactify the moduli space of Higgs bundles on a curve for every reductive group; to initiate a theory of perverse truncation and specialization with application to the algebraic topology of Hitchin systems; to determine, via the Ngo support theorem, the Betti and Hodge numbers of the only known remaining irreducible symplectic manifold for which these are not known; to establish and explain certain remarkable hidden symmetries on the cohomology of the moduli spaces of Higgs bundles.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.