Mathematical Sciences: Commutative Algebra, Algebraic Geometry, and Computational Issues

本研究涉及交换代数、代数几何以及这些领域的计算问题。在交换代数中，主要研究者将研究线性分辨率和Huneke关于自由分辨率结构的问题。在代数几何中，他将考虑一个广义的Castelnuovo猜想，通过射影稳定性的紧化雅可比矩阵族，通过微分几何方法的割线和切线变数的退化，以及带状，K3地毯和超流形。这个项目的计算部分涉及参数系统的准则和计算机代数系统的发展，麦考利。这是代数几何领域的研究，是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，在这些领域以及理论计算机科学和机器人技术中也在寻找应用。

This research is concerned with problems in commutative algebra, algebraic geometry and computational issues in those fields. In commutative algebra, the principal investigator will study linear resolutions and Huneke's question on the structure of free resolutions. In algebraic geometry, he will consider a generalized Castelnuovo conjecture, the family of compactified Jacobians by projective stability, degeneracy of secant and tangent varieties by methods from differential geometry, as well as ribbons, K3 carpets and supermanifolds. The computational parts of this project are concerned with criteria for systems of parameters and development of the computer algebra system, Macaulay. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields, as well as, in theoretical computer science and robotics.