Mathematical Sciences: Algebraic Geometry and Category Theory

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This award is concerned with research in algebraic geometry and category theory. The principal investigator will work on using Chow varieties to obtain desingularizations of quotient varieties by algebraic groups and to construct higher-dimensional generalizations of Grothendieck-Knuidsen moduli spaces of stable punctured curves. He will also develop the formalism of braided monoidal 2-categories and apply it to the study of Zamolodchikov's tetrahedra equations. In addition, he will study Koszul duality for quadratic operads as motivated by graph cohomology; polyhedral syzygies among Steinberg relations in order to get a more formal algebraic approach to higher algebraic K-theory; and the setting for generalizing the Langlands correspondence to 2-dimensional schemes by using a certain generalization of a concept of operad. This is reasearch in the field of algebraic geometry, one of the oldestparts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.