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

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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.