Topics in Algebraic Geometry and Related Areas

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0245528 William E. Fulton There has been exciting recent work involving interactions among Schubert calculus (intersection theory on Grassmannians and related flag varieties), eigenvalue and singular value problems in linear algebra, representation theory, geometric invariant theory and related symplectic geometry. In addition to solving the Horn conjecture about eigenvalues of sums of Hermitian matrices, a variety of related problems have been raised and are being solved. There are quantum analogues of many of these problems, which are part of this project. The project also involves a problem of enumerative geometry, and a more difficult problem of constructing moduli spaces of subvarieties to resolve away multiple components. Three writing projects, on stacks, toric varieties, and curves, have each already led to some new research. The proposed research concerns nine topics, most on the borders of algebraic geometry with other areas. Much of algebraic geometry concerns the geometry of quite general spaces. This research, however, concentrates on a collection of special varieties which have a rich structure that one can hope to analyze in detail. These arise in neighboring areas of mathematics, especially representation theory, but also linear algebra, combinatorics, commutative algebra, and symplectic geometry. For example, although the notion of quantum cohomology -- which came from physics -- make sense in great generality, few spaces have had their quantum cohomology thoroughly worked out, and we plan to expand this knowledge in this project. Three book-writing projects are in progress, two of which should be useful to graduate students in algebraic geometry; the third is aimed at undergraduate mathematics majors.