Fulton The research in this proposal concerns a variety of topics in algebraic geometry and related fields, including: topology of algebraic varieties; degeneracy loci and classical groups; configuration spaces; curves on Calabi-Yau threefolds; compactifications of moduli spaces; intersection theory, Chow groups, and relations with K-theory; p-adic Hodge theory; representation theory, including Langlands dual group; group cohomology and Koszul algebras; conic bundles and Mori fibrations. Algebraic geometry has a long and important history in mathematics, particularly because it interacts with so many other fields of mathematics, such as topology, analysis, algebra, and number theory. It has flourished during the last half of the twentieth century, especially in this country. The recent interactions with physics have brought a new energy to the field, since we have learned that very classical problems - like counting the numbers of plane curves passing through a given set of points - are intimately related to ideas like mirror symmetry in modern physics. This is another example of a problem that seemed to be of purely theoretical interest in mathematics that has turned out to have unexpected connections with another subject, to their mutual benefit. At its simplest and most fundamental level, algebraic geometry studies sets defined by polynomial equations; such sets arise in many problems in computer science and robotics, and there is increasing interaction between algebraic geometry and these fields.
