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Texas A&M University

Events for 04/13/2023 from all calendars

Number Theory Seminar

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Time: 2:30PM - 3:30PM

Location: BLOC 302

Speaker: Matthew Kroesche, Texas A&M University

Title: Low-lying zeros of a family of automorphic forms

Abstract: In their paper “Low-Lying Zeros of Families of L-Functions”, Iwaniec, Luo, and Sarnak calculated the distribution of low-lying zeros of families of modular L-functions in level aspect for test functions with support contained in (−2, 2). In this work, we extend this result to thin (in the p-adic sense) subfamilies of these families, obtained by twisting lower-level modular forms by a character, and investigate what can be said about the support and the lower-order terms in the one-level density in this case.

Mathematical Physics and Harmonic Analysis Seminar

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Time: 4:00PM - 5:00PM

Location: BLOC 302

Speaker: Wilhelm Schlag, Yale

Title: Lyapunov exponents, Schrödinger cocycles, and Avila’s global theory

Abstract: In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years. By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson