Harmonic Analysis is the theory of the Fourier transform operator acting on complex valued functions on various fields, such as, finite fields, p-adic fields, reals and the complex numbers. Interestingly, the Fourier transform is a part of a family of operators, that satisfy relations with respect to one another that can be described by a group symmetry structure, strongly related to the symplectic group. This family is called the Weil representation. The Weil representation serves as a bridge that connects classical harmonic analysis with representation theory, it also gives a powerful fresh perspective about the very nature of the theory. Furthermore, the Weil representation is governed by an object from algebraic geometry, called the geometric Weil representation. The geometric Weil representation, developed by the authors, serves as a bridge that connects between harmonic analysis and algebraic geometry, hence enables to solve analytic problems using modern cohomological techniques. The present project revolves around the following themes: canonical model of the Weil representation and its geometrization, the Weil representation in characteristic two, the Weil representation of symplectic similitudes, and applications to various fields.
This project will enhance our understanding of the representation theoretic and algebraic geometric structures that underlie harmonic analysis and their applications. It will also reveal new perspectives about classical statements from number theory and analysis. The PI and Co-PI have presented their work in numerous classes and seminars over the past years. They are also collaborating with people from other scientific areas, such as engineering and physics, on the applications of their results to questions outside pure mathematics. They strongly believe that their methods and results are of interest to the broader scientific community and has the potential to have radical impact on disciplines outside of mathematics.
