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 or 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 the 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, is 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 the study of string theory.The investigator will explore the fundamental aspects of the general theory as well as important examples through three projects: to develop a more flexible theory of constructible sheaves in the contexts of Morse theory and of algebraic geometry over arbitrary fields of definition; to seek new evidence to the P=W Conjecture--one of the leading open questions in the geometry of Higgs bundles and flat connections on algebraic curves--by importing new geometric results from algebraic geometry over fields of positive characteristic; and to study the structure of moduli spaces of Higgs bundles and flat connections over fields of positive characteristic as critical loci, with applications to the structure and cohomology of these moduli spaces.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.