Vasconcelos, 98-01413 Abstract Vasconcelos This award supports research in commutative algebra, algebraic geometry and computational algebra. It is centered on the development of a new series of numerical signatures of rings and algebras, the so-called homological degrees, that play in local rings and arbitrary algebras a role similar to Castelnuovo--Mumford's regularity in graded structures. It will also emphasize the algebraization (particularly Cohen-Macaulayness and normality) of blowup algebras---Rees algebras of ideals and modules and tangent and secant algebras---and the examination of algebras associated to commuting sets of elements of Lie algebras, the computation of rings of invariants of connected groups and algebras of combinatorial interest. Commutative algebra is broadly concerned with solutions of structured sets of polynomial and analytic equations, and the study of pathways to methods and algorithms that facilitate the efficient processing in large scale computations with such data. These equations often arise internally in the construction of new geometric structures or in domains of applications, such as control and coding theory and robotic motion. A mathematically focused approach is required in order to beat the complexity in problems of interest.
