Algebraic and Analytic Methods in the Mathematical Sciences

这个REU网站以代数和应用分析的同步课程为特色，每个课程都强调计算机辅助的协作实验。分析组件涉及信号的过滤、检测和压缩中的问题。这些问题自然而然地出现在许多现实世界的环境中，而本研究中的学生使用的主要数学方法是小波分析。典型的学生项目包括气象数据的卫星传输，指纹图像的压缩和存储，以及火箭发动机的燃料耗尽。这些调查既需要符号计算，也需要数值计算。代数部分专注于非对易代数方程有限系统的矩阵解和相关的计算问题。这些代数系统通常定义更抽象的结构(例如，群、李代数或其推广)，并在现代数学物理中发挥了重要作用。典型的学生专题集中于量子群和李超代数的不可约表示，以及一般矩阵代数的生成。符号计算在整个过程中扮演着关键的角色。额外的项目将与该部门的NSF Vigre赠款赞助的研究研讨会课程一起组织。

This REU Site features simultaneous programs in algebra and applied analysis, each emphasizing computer-assisted collaborative experimentation. The analytic component is concerned with problems in the filtering, detection, and compression of signals. These questions naturally arise in many real-world settings, and the primary mathematical technique used by the students in this study is wavelet analysis. Typical student projects involve satelite transmission of meteorological data, compression and storage of fingerprint images, and fuel burnout in rocket engines. Both symbolic and numeric computations are needed for these investigations. The algebraic component is focused on matrix solutions to finite systems of noncommutative algebraic equations and on related computational issues. These algebraic systems usually define more abstract structures (e.g., groups, Lie algebras, or generalizations thereof), and have played an important role in modern mathematical physics. Typical student projects focus on the irreducible representations of quantum groups and Lie superalgebras, and on the generation of generic matrix algebras. Symbolic computation plays akey role throughout.Additional projects will be organized in conjunction with research seminar courses sponsored by the department's NSF VIGRE grant.