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1:00pm to 2:00pm - RH 340N - Dynamical Systems Nicholas Chiem - (UC Riverside) Uniformly positive Lyapunov exponents with monotone potentials along local unstable leaves Abstract: Anderson localization is a physical phenomenon that was observed by Phillip Anderson. One definition of localization is when the spectrum of the Schrödinger operator has pure point spectrum with exponentially decaying eigenfunctions. The Lyapunov exponent plays a central role in studying the phenomenon, as uniformly positive Lyapunov exponents paired with a large deviation estimate has been a large indication of localization. Our focus is when one can prove uniform positivity, as Kotani theory would imply that the family of Schrödinger operators has an empty absolutely continuous spectrum. In our talk, we discuss the setting and the methods used to show uniformly positive Lyapunov exponents for lower Hölder potentials along local unstable leaves when generated by hyperbolic dynamics with at least one expanding direction. |
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4:00pm to 5:00pm - RH 306 - Differential Geometry Xiaokang Wang - (UC Irvine) Collapsing Constant Scalar Curvature Metrics In this talk, I will discuss a recent result on collapsing constant scalar curvature (CSC) metrics. We prove that a sequence of CSC metrics that is collapsing with bounded curvature to a manifold can be perturbed to a sequence of nilpotent-invariant collapsing CSC metrics, under a natural assumption involving the eigenvalues of the drift Laplacian on the limiting manifold. This proves a special case of a conjecture of Cheeger-Fukaya-Gromov. We also give some natural conditions on the limiting metric-measure space under which the eigenvalue assumption is automatically satisfied. The upshot is that we build the Cheeger-Fukaya-Gromov theory in the finite regularity setting and solve the perturbed Yamabe equation in the weak setting. This is a joint work with Jeff Viaclovsky. |
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2:00pm to 3:00pm - 340P Rowland Hall - Combinatorics and Probability Grigoris Paouris - (Texas A&M University) Quantitative Versions of Anderson's theorem in Gauss space Anderson's theorem states that for a symmetric log-concave measure on R^n, among all translates of a given symmetric convex set,the measure is maximized at the origin. We develop quantitative counterparts to this result for the Gaussian measure by investigating the quantity: |