This proposal is for a conference that will take place at the University of Michigan, Ann Arbor, July 31--August 5, 2008. The conference will focus on recent advances in commutative algebra, centered around areas which have been greatly influenced by the contributions of Mel Hochster. These include tight closure, singularities and multiplier ideals, the homological conjectures, invariant theory, local cohomology, the structure of free resolutions, and combinatorial commutative algebra. Hochster's research has had a transforming impact on the field, and there are several highly active areas which rely on his past and present work. We hope to bring together these areas for fruitful interaction. Commutative algebra is the study of equations: to paraphrase Hochster, "...algebra, after all, has to do with solving equations. Abstract algebra is the daughter of the theory of equations (in the broadest sense) and perhaps its best theorems still deal with that subject." Striking developments in the last few decades in commutative algebra have reinforced this view of commutative algebra. These developments include solutions to classical problems, as well as strengthened connections with diverse areas including algebraic topology, group cohomology, algebraic combinatorics, representation theory, invariant theory, and algebraic and arithmetic geometry. The invited speakers include experts in different areas, as well as several young researchers. The conference will provide an opportunity to present key results and recent advances. A special effort will be made to support the expenses of young researchers, women and minorities, graduate students, and those without alternative means of support.
