Program

Commutative algebra comes from several sources: the 19th century theory of equations, number theory, invariant theory and algebraic geometry. To study the set of solutions of an equation, e.g. the circle $x^2 + y^2 = 1$ in $C^2$, one can form the ring $C[x,y]/(x^2+y^2-1)$ where $ C$ is the complex numbers. This ring represents polynomial functions on the circle. In a similar manner, to study the zero set of a system of polynomial equations over the complex or real numbers, $$F_1(x_1,...,x_m) = ... = F_n(x_1,...,x_m) = 0,$$ we can form a ring which represents the polynomial functions on this zero set and study its algebraic properties. This investigation can often be reduced to the study of the polynomial functions $F_1,...,F_n$ themselves and techniques of commutative algebra provide significant insight into their properties. When studying the zero set of the $F_i$, it is convenient to study the set of all polynomials $\sum_iF_iG_i$ where the $G_i$ are other polynomials. All such functions share the common zeroes of the $F_i$. The set of all such sums is an \it ideal\rm. A classic development of Kummer's from the 19th century was the realization that ideals could be factored much like numbers can be factored into products of primes. A modern version of this factorization is the powerful tool of primary decomposition of ideals. More recently, Grobner bases have been shown to be extremely useful to analyze properties of such ideals. A significant development over the last 20 years is the role that commutative algebra is taking as a tool for solving problems from a rapidly expanding list of disciplines. In oversimplified terms, the process could be described as follows: Mathematicians with various backgrounds discover ways of encoding information of interest into commutative rings and their modules, then use algebraic concepts, methods, and results to analyze that information efficiently. A famous example of such encoding is the translation of an abstract finite simiplicial complex into an ideal represented by the zeroes of square-free monomials in a set of variables corresponding to the vertices of the simplicial complex. Of course, the existing body of work in commutative algebra is not tailored to suit all new demands. This is precisely where the subject benefits most from the recent surge of external interest, as it receives an influx of novel questions, points of view, and expertise. Our year-long program will highlight these recent developments and will include the following areas: Tight closure and characteristic p methods Toric algebra and geometry Homological algebra Representation theory Singularities and intersection theory Combinatorics and Grobner bases The program will hold an Introductory Workshop in the early fall (dates TBA). Invited speakers include David Benson (University of Georgia), David Eisenbud (MSRI), Mark Haiman (UC Berkeley), Melvin Hochster (University of Michigan), Rob Lazarsfeld (University of Michigan), and Bernard Teissier.