Assistant Professor of Mathematics Djordje Milićević Receives NSA Grant

Posted September 11th, 2014 at 10:54 am.

Assistant Professor of Mathematics Djordje Milićević has received a Young Investigator Grant from the National Security Agency’s Mathematical Sciences Program.  This award is available to promising investigators within ten years after receiving the Ph.D.

Milićević’s proposal is titled “Arithmetic Manifolds, L-Functions, and Analysis”

From the Project Summary:

“This research project centers around two principal themes, that of extremal behavior of high-energy eigenfunctions on arithmetic manifolds, and that of the depth aspect in analytic number theory.

A central question of quantum chaos is the relation between the chaotic dynamics of a classical system, and the asymptotic behavior of its quantum counterpart in the semiclassical limit (inthe case to be studied, between the geodesic flow on a negatively curved manifold and the large eigenvalue limit of the Laplacian eigenfunctions). On certain arithmetic manifolds with a specific geometric and functional structure, the eigenfunctions (Maass forms) exhibit power growth, which is neither generically expected nor predicted by physical models. The ultimate goal of this project is to understand this and related phenomena of concentration of mass on arithmetic manifolds, the precise structure which drives it, and the nature of its place within the correspondence principle framework. Problems to be investigated include exhibiting classes of high-dimensional manifolds saturating the Purity and Subconvexity conjectures, weight distribution of holomorphic modular forms, and topography of arithmetic hyperbolic 3-manifolds including subconvexity and impact of embedded geometric features. Techniques from number theory, spectral analysis, representation theory, random matrix theory and statistics will be employed; in addition, the PI and his students will perform explicit computations to inform general conjectures.”

In number-theoretic problems involving characters and automorphic forms of large level, the depth aspect, which is concerned with highly powerful conductors, plays a very distinctive role. The structural impact of the depth aspect on nonvanishing, subconvexity, and moments of L-functions, as well as exponential sums involving p-adically analytic fluctuations will be studied using p-adic analysis, analytic number theory, and spectral theory. Applications of proposed research on towers of L-functions include choices of moduli for RSA encription.”

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