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Explore advanced research in analytic number theory through this 57-minute conference talk examining the statistical behavior of low-lying zeros in families of L-functions. Delve into the Katz-Sarnak conjecture, which proposes that zero statistics of L-function families correspond to eigenvalue scaling limits from random matrix theory. Learn about recent collaborative research on nth-centered moments of one-level densities for a large orthogonal family of L-functions associated with holomorphic Hecke newforms of level q, averaged over q~Q. Understand how these nth centered moments relate to n-level densities of low-lying zeros and discover verification of the Katz-Sarnak conjecture for these statistics within the range where Fourier transform supports of test functions lie in (-4, 4). Examine key mathematical challenges including identification of off-diagonal main terms and resolution of combinatorial problems that bridge number-theoretic results with random matrix theory predictions. Gain insights into cutting-edge techniques in analytic number theory and the deep connections between L-functions and random matrix theory through this specialized mathematical presentation delivered at the Simons Foundation's Conference on Universal Statistics in Number Theory.
