This project involves four separate directions. The first direction is an investigation of the weak Lefschetz property in positive characteristic, which from the PI's point of view is related to finding the smallest possible degrees of relations on certain polynomials. In positive characteristic, the Frobenius endomorphism plays an important role in producing relations of unexpectedly small degree.The second direction of the project is the study of certain classes of rings for which there are no non-free totally reflexive modules. Among the rings under consideration we mention rings which are almost Gorenstein (in the sense defined by Huneke and the PI in 2003), and determinantal rings. The PI wishes to investigate structural restrictions that are imposed on the ring (such as the Hilbert function) by the existence of a non-free totally reflexive module. The third direction is towards using our knowledge of degrees of relations developed in the first part in order to compute Hilbert-Kunz multiplicities. The fourth part of the project is an attempt to solve a conjecture of Conca, Krattenthaler, and Watanabe regarding when is a certain set consisting of sums of powers of the variables a system of parameters in the polynomial ring with three variables.
The PI works in commutative algebra, a branch of algebra mainly concerned with the study of modules and ideals in a ring. Most of the objects under consideration arise from algebraic geometry, corresponding to sets of solutions of polynomial equations. Homological commutative algebra has to do with finding relations between these solutions, and iterating the process by finding relations on relations, etc., thus giving rise to possibly infinite resolutions. The PI also uses positive characteristic methods in most of her work. In a positive characteristic setting, the ring is endowed with an additional structure called the Frobenius homomorphism. The PI's work takes advantage of this additional structure to show that it gives rise to interesting new phenomena that do not occur in its absence.
