Local Cohomology in Algebra and Geometry

研究者打算研究在代数几何、交换代数和偏微分方程理论的组合方面的相互作用领域中出现的局部同调理论问题。这些问题包括与环和模的结构性质有关的问题，以及旨在获得定量结果的问题。该研究的一个共同环节是局部上同，这是一个将代数几何、d模理论和交换代数联系在一起的概念。研究者将特别探索某些非消失局部模与a -超几何系统解结构的相互作用，并着眼于跳变现象。相关研究旨在了解局部系统沿超平面排列的Bernstein—Sato多项式和跳跃轨迹。在另一个方向上，研究者将研究flyubeznik解决局部上同的有限性质的一个基本问题。这是一个数学领域的计算机辅助项目，称为代数几何。在20世纪，代数几何已经从解析几何转变为一门更加复杂的科学。结果是一个复杂但强大的方法来研究曲线，曲面和其他由多项式定义的几何对象。这种现代的几何方法允许数学家在许多其他情况下使用几何技术和直觉，包括（但不限于）机器人运动规划、计算机视觉、统计和计算机安全。代数几何中使用的方法范围很广。研究者的工作集中在微分学和计算机能力在该学科中的应用，从而将几何、代数、微积分和现代技术结合在他的工作中。随着他继续通过理论和计算手段揭示这些对象之间的相互作用，代数几何作为数学、物理和工程的其他部分的工具变得越来越有价值。

The investigator intends to study problems from the theory of localcohomology that arise in interacting areas of algebraic geometry,commutative algebra, and through combinatorial aspects of the theoryof partial differential equations. These problems include questionsthat relate to structural properties of rings and modules, as well asthose aimed at obtaining quantitative results. A common link to theproposed study is local cohomology, a concept that ties togetheralgebraic geometry, D-module theory, and commutative algebra. Theinvestigator will specifically explore the interaction of certainnon-vanishing local modules with the structure of the solutions ofA-hypergeometric systems, with a view towards jump phenomena. Arelated study is aimed at understanding the Bernstein--Sato polynomialand jump loci of local systems along hyperplane arrangements. In adifferent direction, the investigator will study a basic question ofLyubeznik addressing finiteness properties of local cohomology.This is a computer-aided project in the mathematical area known asalgebraic geometry. During the 20-th century, algebraic geometry haschanged its nature from analytic geometry into a much more complexscience. The result is a complicated but powerful method for studyingcurves, surfaces and other geometric objects defined bypolynomials. This modern approach to geometry allows mathematicians touse geometric techniques and intuition in many other situations,including (but not restricted to) robot motion planning, computervision, statistics, and computer security. The methods used inalgebraic geometry are of a very wide range. The investigator's workconcentrates on the applications of differential calculus and computerpower to the subject, thus combining geometry, algebra, calculus andmodern technology in his work. As he continues to uncover theinterplay of these objects by theoretical and computational means,algebraic geometry is becoming ever more valuable as a tool in otherparts of mathematics, physics and engineering.