Local Cohomology in Algebra and Geometry

调查员打算研究问题从理论的局部同调所产生的相互作用领域的代数几何，交换代数，并通过组合方面的theoryof partial differential equations。 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。一个共同的链接到拟议的研究是局部上同调，一个概念，联系在一起的代数几何，D-模理论和交换代数。研究者将特别探讨某些非零局部模与A-超几何系统解的结构之间的相互作用，并着眼于跳跃现象。相关的研究旨在理解局部系统沿着超平面排列的伯恩斯坦-佐藤多项式和跳跃轨迹。在一个不同的方向，调查员将研究一个基本问题Lyubeznik解决有限性性质的局部上同调。这是一个计算机辅助项目在数学领域被称为代数几何。世纪，代数几何从解析几何转变为一门复杂的科学。其结果是研究曲线、曲面和其它由多项式定义的几何对象的一种复杂而有力的方法。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是非常广泛的。研究者的工作集中在微分学和计算机能力的应用，从而结合几何，代数，微积分和现代技术在他的工作。随着他继续通过理论和计算手段揭示这些对象的相互作用，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.