Research in Commutative Algebra

这个项目涉及最小自由分解的结构及其在代数几何中的应用。私家侦探将就以下主题进行研究： 1)外代数上Hilbert格式的连通性和光滑性（与M. Stillman）; 2)完全交上自由分解的渐近性质和结构; 3)有限幂环上的自由归结（与V. Gasharov和T.所提出的研究是在交换代数和代数几何的密切相关的领域。 这些领域研究多项式方程组的代数不变量和由多项式消失定义的集合的几何。 通常很难明确地找到多项式方程组的解，但是可以研究解集的几何性质。 这些领域的一些成果在通信和机器人技术中有应用。

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.