REU Site: Fractals and Stochastics at UConn

REU 网站：康涅狄格大学的分形和随机指标

• 批准号: 2349433
• 负责人: Luke Rogers
• 金额: $ 46.49万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-05-01 至 2027-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2349433&HistoricalAwards=false
• 关键词: REU; Site; Fractals; Stochastics; UConn

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.

REU网站每年将介绍大约10名本科生的数学研究，让他们参与密集的10周研究项目，通过产生可验证的结果，会议演讲和海报来推进数学各个领域的知识。除了研究经验，学生将练习写作和演示数学作品，学习使用软件工具进行数学研究和出版，获得如何申请研究生院的指导，并通过系列研讨会遇到广泛的数学主题。 学生将主要来自非博士学位授予机构，招聘将强调数学科学中代表性不足的群体，包括妇女和少数民族。一个主要目标是通过增加这些学生继续研究生学习和/或在这些领域工作的可能性，为数学和数学密集型领域的人力资源开发做出贡献。本科研究项目将主要在分形和无序空间的分析方面（分形微分方程，谱抽取方法），随机分析（随机微分和差分方程的遍历性，李雅普诺夫指数），几何分析（离散海森堡群上随机游动的几何性质和与自相似群相关的Schreier图，函数不等式，拉普拉斯坐标中的非欧几里德结构的几何学）。 大多数项目将涉及对重要实例的详细研究，这些实例可以在最小的先决条件下启动，但其结果有望阐明理论的更大部分。这些项目与其他科学领域有着实质性的联系;例如，随机学项目涉及数学物理问题，拉普拉斯特征坐标及其变体广泛用于机器学习，分形和无序介质可用于模拟地质学（多孔岩石中油或水的分布）、计算机科学和神经生物学（神经网络结构）等领域的物理结构。 有关该计划的更多详细信息，请访问https://mathreu.uconn.eduThis奖项反映了NSF的法定使命，并且通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。