REU SITE: Interactions Between Algebra, Computation, and Mathematical Physics

REU 站点：代数、计算和数学物理之间的相互作用

• 批准号: 0138991
• 负责人: Edward Letzter
• 金额: $ 18.09万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-05-15 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0138991&HistoricalAwards=false
• 关键词: REU; SITE; Interactions; Between; Algebra

This REU features simultaneous subprograms in algebra and applied mathematics. The unifying theme is the interplay between algebraic methods, computational techniques, and mathematical physics. Throughout,there is an emphasis on computer-aided experimentation in a collaborative environment.Students in the algebra subprogram learn to develop -- and prove --mathematical conjectures based on extensive studies of specific examples. The primary mathematical focus is on (finite dimensional) algebraicrepresentation theory -- that is, the study of matrix solutions to systems of polynomial equations in noncommuting variables. Investigations along these lines have played, and continue to play, a key role in modern physics. Moreover, solutions to these systems correspond to objects in ``noncommutative geometric spaces.'' The particular systems considered in this subprogram are mostly composed of quadratic and cubic polynomials in two or three (noncommuting) variables. These systems typically define more abstract algebraic objects whose structures can form the basis for deeper investigations. The use of advanced symbolic computation software plays an important role throughout.Students in the applied mathematics subprogram are introduced to the three components of applied mathematics research: modeling, analysis, and computation. The focus is on two topics arising in materials science: complex electro-magnetic permittivity of dielectrics, and exact relations for fiber-reinforced composites. Students working on complex electro-magnetic permittivity develop mathematical stratgies forpredicting how much electro-magnetic energy a material will absorb, in a given frequency, if data for other frequencies is already known; such prediction techniques have the potential to reduce the cost ofexperimental measurements drastically. Students working on exact relations mathematically analyze the influence of composite microstructure on macroscopic properties; understanding this influence is crucial todeveloping new materials with various desired properties -- potentially applicable industrial products include skis, golf clubs, and aircraft parts.

这REU功能同时在代数和应用数学的子程序。 统一的主题是代数方法，计算技术和数学物理之间的相互作用。在整个课程中，重点是在协作环境中进行计算机辅助实验。代数子课程的学生学习在对具体例子进行广泛研究的基础上开发和证明数学公式。 主要的数学重点是（有限维）代数表示论--即研究非交换变量的多项式方程组的矩阵解。沿着这些路线的研究在现代物理学中发挥了并将继续发挥关键作用。此外，这些系统的解对应于“非交换几何空间”中的对象。在这个子程序中考虑的特殊系统主要由两个或三个（非交换）变量的二次和三次多项式组成。这些系统通常定义更抽象的代数对象，其结构可以形成更深入研究的基础。高级符号计算软件的使用在整个过程中起着重要的作用。应用数学子程序的学生被介绍给应用数学研究的三个组成部分：建模，分析和计算。重点是在材料科学中出现的两个主题：复电磁介电常数，和纤维增强复合材料的精确关系。研究复杂电磁介电常数的学生开发数学策略，用于预测在给定频率下，如果其他频率的数据已经知道，材料将吸收多少电磁能量;这种预测技术有可能大幅降低实验测量的成本。研究精确关系的学生通过数学分析复合材料微观结构对宏观性能的影响;理解这种影响对于开发具有各种所需性能的新材料至关重要-潜在的适用工业产品包括滑雪板，高尔夫球杆和飞机部件。