Studies in Commutative Algebra

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The investigator will work on some questions in the theory of commutative rings that arise from tight closure theory, the theory of intersection multiplicities, a study of local cohomology modules, and from the homological conjectures for local rings. In tight closure theory, there is a substantial focus on understanding the class of F-regular rings and on developing theorems for this class of rings. The study of local cohomology modules addresses finiteness issues of these modules over regular rings of mixed characteristic, and another question that is related to the theory of solid closure. Research on Hilbert-Kunz multiplicities and on the rigidity of the Tor functor will be carried out in continued collaboration with Claudia Miller. Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are certain functions on these solution sets. The connections between algebra and geometry are largely due to the revolutionary work of Grothendieck, Serre and Zariski. An active area of research in commutative algebra today is the theory of tight closure developed by Hochster and Huneke. This theory provides stronger formulations of existing theorems, and brings together many seemingly unrelated problems. It also has strong connections with the study of singularities of geometric objects. The theory of local cohomology was developed by Grothendieck who used it to obtain several striking results. Local cohomology has applications to basic, and yet deep, questions such as determining the minimal number of polynomial equations needed to define an algebraic set. It continues to develop a fascinating interaction with several other branches of mathematics.