William Haboush - US grants

This project is concerned with several problems in commutative algera. The principal investigators will work on the Horrocks'conjecture. They will also work on the syzygy theorem and extending certain results to the mixed characteristic case. Another area of concentration concerns invariants related to representations of groups in positive characteristic. One of the main problems to be considered in this area is to describe how tensor functors, acting on indecomposable representations, decompose. The research in this project involves two areas of commutative algebra. The first area is concerned with the structure and properties of syzygies of finite projective dimension and uses local commutative algebra. A "syzygy" is a linking or yoking together of steps in a projective resolution describing the structure of a module. The second area is concerned with decomposing tensor functors under formal group laws and the properties of the relative Grothendieck groups that are associated to the decomposition.

This research is in the area of group representation theory. The principal investigator will develop an algebraic harmonic analysis to transfer the representation theory of semisimple groups to the representation theory of Hecke algebras. The central problem to be considered in this project is the Lusztig conjecture which states that certain invariants attached to an algebraic group can be calculated from polynomials. This is a difficult and central problem in group theory. It also has implications in other areas of mathematics.

In the last twenty years, few areas of mathematics have developed as extensively as the theory of algebraic groups. Recent developments in the theory of infinite dimensional Lie algebras, in noncommutative algebraic geometric analogues of algebraic groups such as quantum groups, and in representation theory, suggest that an overall assessment of recent progress might be in order. This grant will support the 1991 Summer Research Institute on Algebraic Groups and their Generalizations, to be held at Pennsylvania State University, July 8-28, 1991. The three week institute is the thirty-ninth in a series sponsored by the American Mathematical Society which brings together groups of mathematicians interested in a particular field of mathematical research. Emphasis is placed on instruction at a very high level with seminars and lectures by distinguished mathematicians in related fields, to promote interaction between participants while broadening their mathematical perspectives.