The principal investigator plans to pursue two projects devoted to the study of interactions between random matrix theory, number theory and combinatorics. In the first project, building on his recent joint work with Persi Diaconis and with Brian Conrey, the principal investigator will explore connections between the distribution of the secular coefficients of random matrices, the conjectures of Conrey, Farmer, Keating, Rubinstein and Snaith for moments of L-functions, and some classical problems in enumerative combinatorics related to counting magic squares. The second project of the principal investigator is devoted to studying from a unified point of view one of the main problems in the theory of expander graphs and one of the basic conjectures in the theory of quantum chaos. A basic problem in the theory of expander graphs, formulated by Lubotzky and Weiss, is to what extent being an expander family for a family of Cayley graphs is a property of the groups alone, independent of the choice of generators. For many natural families of groups, in particular, for special linear group of order two, numerical experiments indicate that it might be an expander family for generic choices of generators (Independence Conjecture). A basic conjecture in Quantum Chaos, formulated by Bohigas, Giannoni, and Shmit, asserts that the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of a typical member of the appropriate ensemble of random matrices. Both conjectures can be viewed as asserting that a deterministically constructed spectrum generically behaves like the spectrum of a large random matrix: in the bulk (Quantum Chaos Conjecture) and at the edge of the spectrum (Independence Conjecture). The principal investigator will work on proving these conjectures in the context of the spectra of elements in group rings.
Random Matrix Theory originated in Wigner's suggestion in the early fifties that the resonance lines of heavy nuclei might be modelled by the spectrum of a large random matrix. In the ensuing fifty years the scope and depth of Random Matrix Theory has dramatically increased; in the past decade the subject has undergone explosive growth. The first project of the principal investigator aims at forging a link between two recent lines of development in Random Matrix Theory. One is the discovery and exploitation of the connections between eigenvalue statistics and the longest-increasing subsequence problems in enumerative combinatorics; another is the outburst of interest in characteristic polynomials of random matrices and associated global statistics, particularly in relation with the moments of the Riemann zeta function, a function of fundamental importance in number theory. The second project of the principal investigator is devoted to studying connections between random matrices and expander graphs -- highly connected sparse graphs that efficiently propagate information quickly to many nodes along short paths. Recent explicit constructions of such graphs have created an explosion of interest in their potential applications to network design, complexity theory, coding theory and cryptography. These constructions are algebraic in nature, and provide a beautiful example of how abstract, seemingly unrelated topics in number theory, group theory and combinatorics can be elegantly combined to solve an important real-world problem.
