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

数学科学：交换代数与分析、几何和计算机科学的相互作用

• 批准号: 9625308
• 负责人: Karen Smith
• 金额: $ 20万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625308&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Interactions; Commutative; Algebras

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.

小行星9625308 职业发展计划将交换代数研究项目与密歇根大学研究生和本科生的教育项目相结合。 大多数拟议的研究是一个更广泛的计划的一部分，以阐明分析思想之间的联系，如微分算子和L^2估计与交换代数中的特征p技术，如紧闭包。本论文的研究目标是：（1）应用特征p技巧解决代数几何中的Fujita猜想，（2）理解复簇上线丛的上同调在紧闭包下的消失定理，（3）利用特征p方法研究非光滑簇上微分算子的环论性质，（4）利用特征p方法研究非光滑簇上微分算子的环论性质，（5）利用特征p方法研究非光滑簇上微分算子的环论性质，（6）利用特征p方法研究非光滑簇上微分算子的环论性质，（7）利用特征p方法研究非光滑簇上微分算子的环论性质，（8）利用特征p方法研究非光滑簇上微分算子的环论性质。以及（4）研究$D$-模块和具有Frobenius动作的模块之间似乎存在直接联系的特定实例。职业发展计划的第一和第二部分概述了进行研究的具体方法。主要工具是理想上的闭包运算，如紧闭包和积分闭包，以及局部上同调的Frobenius作用。提出了一个适合研究生的计算机项目，作为理解微分算子环上的环的模结构的方法。 第三部分提出了一个四年计划，教研究生和本科生投影代数几何和计算机代数，同时从事他们的国家的最先进的研究。 拟议的项目是一个大大扩展的版本已经在进行中的稀疏系统的参数和Noether复杂性的研究。 这个长期的研究和教育项目包括计划在密歇根大学的一个新的研究生课程强调使用计算机在代数几何，并为初级研讨会的专业在数学和计算机科学团队的本科生领导的研究生学习射影几何通过自我发现。 在这次研讨会开始的研究将扩大到一个全面的计算机代数研究项目。 该项目将在一次会议上达到高潮，学生们将向更广泛的受众展示他们的发现。 第四部分论述了体育教师职业目标的教育和人力资源方面。提出了一个将商业和工业中的数学实践者带到本科课堂的计划。目标是同时激励课程，并向学生介绍像他们一样使用数学创造成功职业的人。