RTG: Number Theory and Representation Theory at the University of Michigan

该奖项将支持密歇根大学数论和表示论的研究培训小组。RTG将围绕这些领域的教师建立一个研究小组，支持博士后，研究生和本科生。一个由八名教师组成的团队（Bhargav Bhatt，Stephen DeBacker，Wei Ho，Tasho Kaletha，Jeffrey Lagarias，Kartik Prasanna，Andrew Snowden和Michael Zieve）将监督该项目。该项目将支持一些倡议，以扩大在这些数学领域的参与，并鼓励新的合作模式。这些举措包括：㈠三人小组合作，其中至少有三名参与者组成的纵向一体化团队共同研究项目; ㈡一系列本科生计算倡议讲习班，通过这些讲习班，本科生将通过研究涉及理论和编码混合的计算问题来进行数学研究;（iii）一系列不同形式的夏季研讨会，将涉及并受益于来自全国各地的年轻数学家;（iv）一个更传统的REU计划;（v）一个涉及密歇根州其他大学的“数论日”。将有几个新的招聘举措，将包括桥梁硕士课程的扩展和本科课程的发展，这将普及本提案中的数学，并使其更容易获得密歇根大学本科人口中的不同受众。数论和表示论都是数学的核心领域，并且高度相互关联。这些领域之间的联系在朗兰兹纲领中最为明显，该纲领预测了数域上多项式的根与代数群的表示之间的关系;这种联系是数学中一些最引人注目的成就的原因，例如费马最后定理的证明。在这些领域培养下一代数学家对国家至关重要，因为他们在数学中发挥着核心作用，也因为这些领域在密码学、加密货币和区块链技术等领域的重要实际应用。近年来，该领域有了重大的发展，包括完美空间理论、自守表示的分类、数的几何和算术不变理论以及代数圈的算术理论。该基金会的PI们拥有涵盖所有这些最新发展的专业知识，并将通过培训博士后、研究生和本科生来沿着这些知识，以增加在这些领域从事职业的美国公民的数量。该奖项反映了NSF的法定使命，并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。

This award will support a Research Training Group in Number theory and Representation theory at the University of Michigan. The RTG will build a research group around the faculty in these areas, supporting postdocs, graduate students and undergraduates. A team of eight faculty members (Bhargav Bhatt, Stephen DeBacker, Wei Ho, Tasho Kaletha, Jeffrey Lagarias, Kartik Prasanna, Andrew Snowden, and Michael Zieve) will oversee the project. The project will support a number of initiatives to broaden participation in these areas of mathematics, and to encourage new modes of collaboration. These initiatives include among others (i) the Teams of Three collaborations, in which vertically integrated teams of at least three participants work jointly on research projects; (ii) a series of Undergraduate Computational Initiative Workshops, through which undergraduates will be introduced to research in mathematics by working on computational problems that involve a mix of theory and coding; (iii) a series of summer workshops with varying formats that will involve and benefit young mathematicians from across the country; (iv) a more traditional REU program; and (v) a "Number theory day" involving other universities in Michigan. There will be several new recruitment initiatives that will include an expansion of a bridge Masters program and the development of undergraduate classes that will popularize the mathematics in this proposal and make it more accessible to diverse audiences within the undergraduate population at Michigan. Number theory and Representation theory are both central areas in mathematics, and are highly interconnected. The connection between these areas is most visible in the Langlands program, which predicts relations between the roots of polynomials over number fields and the representations of algebraic groups; this connection is responsible for some of the most striking achievements in mathematics, such as the proof of Fermat's last theorem. Training the next generation of mathematicians in these areas is of vital importance to the country both because of the central role that they play in mathematics and because of important practical applications of these areas to topics such as cryptography, cryptocurrencies and blockchain technology. In recent years, there have been major developments in the field, including the theory of perfectoid spaces, the classification of automorphic representations, the geometry of numbers and arithmetic invariant theory, and the arithmetic theory of algebraic cycles. The PIs on this grant together have expertise covering all of these recent developments and will pass this along by training postdocs, graduate students and undergraduates, with the goal of increasing the number of US citizens pursuing careers in these areas.This 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.