REU Site: Fractals and Stochastics at UConn

REU网站每年将向大约10名本科生介绍数学研究，让他们参与为期10周的密集研究项目，通过产生可发表的结果、会议演讲和海报来促进数学各个领域的知识。除了研究经验，学生还将练习写作和展示数学作品，学习使用软件工具进行数学研究和出版，接受如何申请研究生院的指导，并通过一系列研讨会遇到广泛的数学主题。学生将主要来自非授予博士学位的机构，招生将强调数学科学中代表性不足的群体，包括妇女和少数群体。一个主要的目标是通过增加这些学生在这些领域继续研究生学习和/或职业生涯的可能性来促进数学和数学密集领域的人力资源的发展。本科生的研究项目将主要集中在以下方面：分形和无序空间的分析(分形微分方程、谱抽取方法)、随机分析(随机微分方程和差分方程的遍历性质、李雅普诺夫指数)、几何分析(离散海森堡群上的随机游动的几何性质和与自相似群有关的Schreier图、泛函不等式、拉普拉斯坐标下的非欧几里得结构的几何)。大多数项目将涉及对重要例子的详细研究，这些例子可以在最低限度的先决条件下启动，但其结果预计将阐明理论的更大部分。这些项目与其他科学领域有很大的联系；例如，随机项目与数学物理中的问题有关，拉普拉斯特征坐标及其变体被广泛用于机器学习，分形图和无序介质可用于模拟从地质学(多孔岩石中的石油或水的分布)到计算机科学和神经生物学(神经网络结构)等领域研究的物理结构。有关该计划的更多细节可在https://mathreu.uconn.eduThis网站上获得，该奖项反映了美国国家科学基金会的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The REU site will introduce approximately 10 undergraduate students each year to mathematical research by involving them in intensive 10-week research projects that advance knowledge in various areas of mathematics by producing publishable results, conference talks and posters. In addition to research experience, students will practice writing and presenting mathematical work, learn to use software tools for both mathematics research and publication, receive guidance on how to apply to graduate school, and encounter a broad range of mathematical topics through a seminar series. Students will be drawn primarily from non-PhD granting institutions and recruitment will emphasize under-represented groups in the mathematical sciences, including women and minorities. A major goal is to contribute to the development of human resources in mathematics and mathematically intensive areas by increasing the likelihood that these students go on to graduate study and/or a career in these fields.Undergraduate research projects will primarily be in aspects of the analysis of fractals and disordered spaces (fractal differential equations, spectral decimation methods), stochastic analysis (ergodic properties of random differential and difference equations, Lyapunov exponents), geometric analysis (geometric properties of random walks on the discrete Heisenberg group and Schreier graphs associated to self-similar groups, functional inequalities, geometry of non-Euclidean structures in Laplacian coordinates). Most projects will involve detailed study of important examples that can be initiated with minimal prerequisites but for which the results are expected to illuminate some larger part of the theory. The projects have substantial connections to other areas of science; for example, the stochastics projects are related to problems in mathematical physics, Laplacian eigencoordinates and their variants are widely used in machine learning, and fractals and disordered media can be used to model physical structures studied in areas from geology (distribution of oil or water in porous rock) to computer science and neurobiology (structure of neural networks). Further details about the program are available at https://mathreu.uconn.eduThis award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.