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

这一奖项将支持密歇根大学数论和表象理论的研究培训小组。RTG将围绕这些领域的教职员工建立一个研究小组，支持博士后、研究生和本科生。一个由八名教职员工(Bhargav Bhat、Stephen DeBacker、魏浩、Tasho Kaletha、Jeffrey Lagarias、Kartik Prasanna、Andrew Snowden和Michael Zieve)组成的团队将监督该项目。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(I)三个合作小组，其中至少有三个参与者组成垂直整合的团队，共同从事研究项目；(Ii)一系列本科生计算倡议讲习班，通过这些讲习班，本科生将被介绍通过研究涉及理论和编码的计算问题来进行数学研究；(Iii)一系列形式多样的暑期讲习班，将有来自全国各地的年轻数学家参与并受益；(Iv)更传统的REU计划；以及(V)有密歇根州其他大学参加的“数字理论日”。将有几项新的招生举措，其中包括扩大桥梁硕士课程，并开发本科课程，普及这项提议中的数学，并使密歇根大学本科生中的不同受众更容易接触到它。数论和表示论都是数学中的中心领域，并且高度相互联系。这些区域之间的联系在朗兰兹程序中最为明显，该程序预测了数域上多项式的根与代数群的表示之间的关系；这种联系促成了一些最引人注目的数学成就，例如费马大定理的证明。培训下一代数学家在这些领域对国家至关重要，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主题上的重要实际应用。近年来，该领域取得了重大进展，包括完备空间理论、自同构表示的分类、数的几何和算术不变理论以及代数圈的算术理论。这笔拨款的私人投资机构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、研究生和本科生来传递这一点，目标是增加在这些领域寻求职业生涯的美国公民的数量。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.