Research Summary
نویسنده
چکیده
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 F where F is a eld which can be either a nite eld or an in nite eld such as the elds 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 nite eld setting and the theory of algebraic Dmodules 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 S 1 R : 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 mathematical physics. In particular, I have obtained several applications of the abstract 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.
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تاریخ انتشار 2009