Mathematical Sciences: Problems in Analysis, Probability, and Finite Mathematics

六名本科生将获得本本科生研究经验奖的支持。他们将在几位教师顾问的指导下工作，他们都有很强的研究项目正在进行中。这个项目延续了华盛顿大学一个成功的本科研究项目。在接下来的两年里，顶尖的数学研究人员将进行一系列的访问，从而扩大该项目。华盛顿大学为这项活动提供了特别丰富的环境，在过去的十年里，华盛顿大学培养了许多进入普特南考试决赛的顶尖选手和领先的团队。在许多可供学生做专题的课题中，包括了对phi变换的研究。这个概念是小波变换的最新变体，它允许将一般函数分解为简单项的和。它具有有趣的计算特性，允许在不需要广泛的理论训练的情况下，早期进行简单但重要的问题。本文还将对共形映射和调和测度的边界行为进行研究。这项工作将很好地适应学生的发展谁最近完成了复杂变量的本科课程。另一个感兴趣的领域包括在几个复变量理论中出现的问题。在这里，研究人员可以构建自同构群的数值模型，并计算与边界微分不变量相关的不变量。这项工作只需要基本的代数和对符号操作程序的理解。在此过程中，学生将发展对几何结构和相关分析的具体理解。一个更实用的程序开发分子进化的数学模型也将可用。学生将考虑从随机交配群体中选择的个体的编码区域中核苷酸分布的问题。与算法的联系，估计系统发育的生物，其特征是已知的，在目前将进行研究。

Six undergraduates will receive support from this Research Experiences for Undergraduates award. They will work under the direction of several faculty advisors all of whom have strong research programs under way. This program continues a successful program of undergraduate research at Washington University. It will be augmented during the next two years by a series of visits by leading mathematical researchers. The environment for this activity is particularly fecund at Washington University which has produced many of the top finalists and leading teams in the Putnam examinations over the past ten years. Included among the many topics which will be available for student projects is the study of the phi-transform. This concept is a very recent variation of the wavelet transform allowing one to decompose general functions into sums of simple terms. It has interesting computational characteristics which allow for simple but important questions to be undertaken early without need for extensive theoretical training. Work will also be done on the boundary behavior of conformal mappings and harmonic measure. This work will fit well into the development of students who have recently completed an undergraduate complex variables course. Another area of interest includes problems arising in the theory of several complex variables. Here, researchers can construct numerical models for automorphism groups and perform calculations of invariants associated with boundary differential invariants. The work only requires basic algebra and an understanding of symbol manipulation programs. In the process, students will develop concrete understanding of geometric structures and the analysis related to them. A more applied program developing mathematical models of molecular evolution will also be available. Students will consider questions concerning the distribution of nucleotides in the coding regions of individuals chosen from a random-mating population. Connections with algorithms for estimating phylogenies of creatures whose traits are known in the present will be investigated.