Mathematical Sciences: Interactions of Commutative Algebras with Analysis, Geometry and Computer Science

9625308史密斯职业发展计划将交换代数的研究计划与密歇根大学研究生和本科生的教育计划结合在一起。大多数拟议的研究是一个更广泛的计划的一部分，目的是阐明微分算子等分析思想与$L^2$估计之间的联系--这些估计具有交换代数中特有的$p$技巧，如紧闭包。这项研究的目的是：(1)应用特征$p$技巧来解决代数几何中Fujita猜想的问题；(2)用紧闭包的形式理解复数簇上线丛的上同调的消失定理；(3)利用归约为特征的$p$方法研究非光滑簇上微分算子的环论性质；(4)研究$D$-模与Frobenius作用下的模之间似乎有直接联系的特殊情况.职业发展计划的第(一)和第(二)部分概述了开展研究的具体方法。主要工具是关于理想的闭包运算，如紧闭包和积分闭包，以及关于局部上同调的Frobenius作用。提出了一个适合研究生的计算机项目，作为一种理解微分算子环上环的模结构的方法。第三部分提出了一个四年计划，教授研究生和本科生射影代数几何和计算机代数，同时让他们从事最先进的研究。拟议的项目是对稀疏参数系统和Noether复杂性的研究的极大扩展。这个长期的研究和教育项目包括计划在密歇根大学开设一门新的研究生课程，强调在代数几何中使用计算机，以及为数学和计算机科学专业的学生举办初级研讨会，在该研讨会上，由一名研究生领导的本科生团队通过自我发现学习射影几何。本次研讨会开始的研究将扩展到计算机代数的全面研究项目。这个项目将在一次会议上达到高潮，在会上，学生们向更广泛的观众展示他们的发现。第四部分论述个人投资促进机构职业目标的教育和人力资源方面。一项将商业和工业中真实的数学实践者带到本科生课堂的计划被提出。目标是同时激励课程，并向学生介绍像他们这样的人，他们利用数学创造了成功的职业生涯。

9625308 Smith The Career Development Plan combines a research program in commutative algebra with an educational program for graduate and undergraduate students at the University of Michigan. Most of the proposed research is part of a broader program to illuminate the connections between analytic ideas such as differential operators and $L^2$-estimates with characteristic $p$ techniques in commutative algebra such as tight closure. The goals of the proposed research are: (1) to apply characteristic $p$ techniques towards a solution of Fujita's Conjecture in algebraic geometry; (2) to understand vanishing theorems for cohomology of line bundles on complex varieties in terms of tight closure; (3) to study the ring theoretic properties of differential operators on non-smooth varieties by using reduction to characteristic $p$ methods; and (4) to study a particular instance where there appears to be a direct link between $D$-modules and modules with an action of Frobenius. Parts (I) and (II) of the Career Development Plan outline specific methods by which the research will be conducted. The main tools are closure operations on ideals, such as the tight closure and the integral closure, and the Frobenius action on local cohomology. A computer project suitable for a graduate student is proposed as a method for understanding the module structure of a ring over its ring of differential operators. Part (III) proposes a four year plan to teach both graduate and undergraduate students projective algebraic geometry and computer algebra while engaging them in state of the art research. The proposed project is a greatly expanded version of research already in progress on sparse systems of parameters and Noether Complexity. This long term research and educational project includes plans for a new graduate course at the University of Michigan stressing the use of computers in algebraic geometry, and for a junior level seminar for majors in mathematics and computer scien ce in which teams of undergraduate students led by a graduate student learn projective geometry through self-discovery. The research begun in this seminar will be expanded to a full scale research project on computer algebra. This project will culminate in a conference in which students present their findings to a broader audience. Part (IV) addresses the educational and human resources aspects of thePIs career goals. A program to bring real-life practitioners of mathematics in business and industry to the undergraduate classroom is proposed. The objectives are to simultaneously motivate the curriculum and introduce students to people like themselves who have used mathematics to create a successful career.