CAREER: Crossroads of Representation Theory and Number Theory

This project will pursue several interdisciplinary research problems at the intersection of number theory, which studies properties of number systems and their interconnections, and representation theory, which is about the linearization of symmetries. The past work of the PI has tackled major conjectures concerning the interaction of these two fields, such as the Breuil–Mezard Conjecture and the Local Langlands Conjecture.

The PI will continue to push these directions, as well as new ones stemming from the attendant technological developments. Furthermore, the project will integrate these pursuits with significant outreach, training, and mentorship efforts encompassing students across all tiers of education from primary to undergraduate to graduate. For example, PI will continue to organize outreach to local schools, research experiences for undergraduates, and graduate training workshops and mentorship opportunities.

Some of the specific research problems are as follows. The PI will further develop his new approach with Le Hung to the Breuil–Mezard Conjecture, which is based on an analogy with mirror symmetry, in order to cover general groups and fields. The PI will also extend the higher Siegel–Weil formula, proved for non-singular terms in joint work with Yun–Zhang, to more general terms, with the eventual goal of completing a higher version of the Kudla program over function fields.

In another direction, the PI will geometrize the classical theory of the Weil representation, and the local theta correspondence, for p-adic groups. These have applications to the categorical Local Langlands correspondence, especially towards Langlands functoriality and the explicit construction of eigensheaves. The PI will supervise and mentor several Ph.D. students and postdocs on projects within these themes.

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.