Minimal Representations and Theta Correspondence

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

This grant will provide support for U.S. based participants to attend a five day workshop titled "Minimal Representations and Theta Correspondence," to be held at the Erwin Schrödinger Institute (ESI) in Vienna, Austria. The event will take place April 11-15 2022. The minimal representations topics discussed at the conference have played a central and unifying role in mathematics for over fifty years, where they find applications to quantum mechanics and communication theory, and within mathematics to number theory. Theta correspondences use minimal representations to connect disparate mathematical objects. Work on theta correspondences has accelerated recently, with the full proof of Howe's Duality Conjecture published just a few years ago. The award will allow U.S. scholars to continue working on minimal representations and theta correspondences on the global stage, with an international community of researchers, as well as introduce a new generation of mathematicians to the material. Minimal representations have applications to automorphic forms, harmonic analysis, for example, the Fourier transforms on basic affine spaces, and mathematical physics, for example, scattering amplitudes in string theory. They originate in the Weil representation of the metaplectic group, where they are connected to Schrödinger's formulation of quantum mechanics via wave functions. Within mathematics, the most profound applications of minimal representations are through theta correspondences -- a generalization of Howe's duality correspondence. Newer "exceptional" theta correspondences have been vital in the Langlands program and arithmetic. This conference is a timely event, bringing participants up-to-date on recent breakthroughs on minimal representations and theta correspondences. The ESI workshop will bring together the top scholars from around the world to showcase the latest progress in this field of mathematics and the grant will allow U.S. scholars to travel to Austria and attend this important event. For more information on the conference, see https://www.esi.ac.at/events/e420. 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.