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

6名本科生将从这个本科生研究体验奖中获得支持。他们将在几位教员顾问的指导下工作，这些人都有强大的研究计划正在进行中。该项目延续了华盛顿大学成功的本科生研究项目。在接下来的两年里，领先的数学研究人员将进行一系列访问，从而扩大这一领域。这项活动的环境在华盛顿大学特别丰富，在过去的十年里，华盛顿大学在Putnam考试中培养了许多顶尖的决赛选手和领先的团队。在学生专题的众多主题中，包括了对Phi变换的研究。这个概念是小波变换的最新变体，它允许人们将一般函数分解成简单项的和。它具有有趣的计算特性，允许在不需要广泛的理论培训的情况下及早解决简单但重要的问题。此外，还将研究共形映射和调和测度的边界性质。这项工作将非常适合刚刚完成本科生复变量课程的学生的发展。另一个令人感兴趣的领域包括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，并执行与边界微分不变量相关的不变量的计算。这项工作只需要基本的代数和对符号处理程序的理解。在这个过程中，学生将发展对几何结构和相关分析的具体理解。开发分子进化数学模型的一个更实用的程序也将可用。学生将考虑从随机交配群体中选择的个体的编码区中核苷酸分布的问题。将研究与用于估计目前已知其特征的生物的系统发育的算法的联系。

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.