Research in Commutative Algebra

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This project deals with the structure of minimal free resolutions and their applications in algebraic geometry. The P.I. will conduct research on the following topics : 1) connectedness and smoothness of Hilbert schemes over exterior algebras (joint with M. Stillman); 2) the asymptotic properties and structure of free resolutions over complete intersections; 3) free resolutions over rings with restricted powers (joint with V. Gasharov and T. Hibi).The proposed research is in the closely related fields of commutative algebra and algebraic geometry. These fields study the algebraic invariants of systems of polynomial equations and the geometry of sets defined by the vanishing of polynomials. Usually it is very difficult to find explicitly the solutions of a system of polynomial equations; however one can study the geometric properties of the set of solutions. Some of the results in these fields have applications in communications and robotics.