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

本研究涉及的问题是交换 代数，代数几何和计算问题， 领域的 在交换代数中，首席研究员将 研究线性归结和Huneke关于结构的问题 自由决议。 在代数几何中，他将考虑一个 广义Castelnuovo猜想，紧化族 雅可比矩阵的射影稳定性、割线的退化性和 切品种的方法，从微分几何，以及 像丝带K3地毯和超流形 计算 该项目的部分内容涉及系统的标准， 参数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， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。 在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。 目前，该领域 不仅使用代数方法，而且使用分析方法， 拓扑学，反过来也在这些领域中找到了应用， 以及理论计算机科学和机器人技术。

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.