RESEARCH SUMMARY

My research concerns the study of the symmetries and the geometry of the basic Hilbert spaces that appear in harmonic analysis and applied mathematics, i.e., the Hilbert spaces of functions (or more generally sections of certain vector bundels) on geometric spaces. For example when I study the Hilbert spaces of functions on vector spaces of the form FN where F is a \u85eld which can be either a \u85nite \u85eld or an in\u85nite \u85eld such as the \u85elds of real or complex numbers I use two intertwining points of view: group representation theory and algebraic geometry. The group representation theory is the HeisenbergWeil representation of the Heisenberg group H(2N;F) and the symplectic group Sp(2N;F): The algebraic geometry is the Grothendieck theory of `-adic sheaves in the \u85nite \u85eld setting and the theory of algebraic D-modules in the real and complex cases. Other examples which arise naturally in my research, are the Hilbert spaces of global section of certain vector bundels on the (N 1)-dimensional sphere SN1 RN: Now, the harmonic analysis tools that I use in the study of these Hilbert spaces are the representation theory of the orthogonal group O(N) and the related di¤erential geometry. My motivation to study these objects comes from the fact that these Hilbert spaces appear naturally in concrete problems of applied mathematics and mathe-theory to discrete harmonic analysis, digital signal processing, arithmetic quantum chaos, and three-dimensional structuring of molecules. This summary shall describe some of my works in pure and applied mathematics. Pure Mathematics 1. The Weil representation over finite fields The discrete Fourier transform (DFT for short) is in a natural way a member of a group of unitary operators that act on the Hilbert space H = C(F) of complex valued functions on the \u85nite \u85eld F = Fp, where p is an odd prime number: The DFT operator satis\u85es the following system w of p2 linear equation: DFT (t; w) = (w (t; w)) DFT; for every t; w 2 F; where w