This project supports fundamental and computational research in algebra, number theory and algebraic geometry. One of the principal investigators will study algebraic K-theory with a particular emphasis on determining the structure of group rings. Another principal investigator will work on extending the notion of motivic cohomology to arbitrary weights which is a significant problem in algebraic geometry. Work will also be done on the theory of p-adic fields. Problems concerning the extension of Buchberger theory and algorithms to rings which have complementary valuation rings will be considered. In addition, this project includes research on the relationship between projective varieties and syzygies, a study of Castelnuovo-Mumford regularity and its relationship to the complexity of standard Groebner bases and syzygies. This project involves research on a wide variety of problems in algebra, algebraic computation and number theory. Although the research covers a wide breath of topics, there are large overlaps and interactions among the investigators on this project. This typifies a growing trend in mathematics of interaction with many diverse areas along with the use and study of computation.
