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

这个REU的特点是代数和应用数学的子程序同时进行。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。整个课程强调在协作环境中的计算机辅助实验。代数子程序的学生学习发展--并证明--基于对具体例子的广泛研究的数学猜想。主要的数学焦点是(有限维)代数表示理论--即研究非对易变量的多项式方程组的矩阵解。沿着这些思路进行的研究在现代物理学中一直并将继续发挥关键作用。此外，这些系统的解对应于“非对易几何空间”中的对象。该子程序中考虑的特殊系统大多由两个或三个(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发挥着重要的作用。应用数学子程序的学生被介绍到应用数学研究的三个组成部分：建模、分析和计算。重点是材料科学中出现的两个问题：介质的复电磁介电常数和纤维增强复合材料的精确关系。从事复杂电磁介电常数研究的学生开发数学策略，在已知其他频率的数据的情况下，预测一种材料在给定频率下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用数学方法分析复合微结构对宏观性能的影响；了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、高尔夫球杆和飞机零部件。

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.