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Alex Blumenthal, University of Maryland College Alex Blumenthal, University of Maryland CollegeEwing 336<p>Title: Random perturbations of predominantly hyperbolic maps </p> <br></br> Abstract: For differentiable dynamical systems, Lyapunov exponents (LE) characterize sensitivity with respect to initial conditions. Estimating LE can be extremely difficult, even for systems which appear, based on numerics, to be quite chaotic: these difficulties are exemplified by the Chirikov standard map, which is strongly hyperbolic on a 'large' but noninvariant subset of phase space. I will discuss recent results demonstrating that in the presence noise with sufficiently large (yet still quite small) amplitude, the estimation of LE can be greatly simplified for a class of "prototypical" 2D maps which are predominantly hyperbolic, including the Chirikov standard map. This is not unreasonable: the real world is inherently noisy, and so it is natural to consider randomly perturbed dynamics. This work is joint with Lai-Sang Young and Jinxin Xue. 10/17/2017 7:30:00 PM10/17/2017 8:30:00 PMFalse
Hyunchul Park, SUNY New PaltzHyunchul Park, SUNY New PaltzEwing 336Title: Spectral heat content for Lévy processes. Abstract: In this talk, we study a short time asymptotic behavior of spectral heat content for Lévy processes. The spectral heat content of a domain D can be interpreted as the amount of heat if the initial temperature on D is 1 and temperature outside D is identically 0 and the motion of heat particle is governed by underlying Lévy processes. We study spectral heat content for arbitrary open sets with finite Lebesgue measure in a real line under some growth condition on the characteristic exponents of the Lévy processes. We observe that the behavior is very different from the classical heat content for Brownian motions. We also study the spectral heat content in general dimensions when the processes are of bounded variation. Finally we prove that asymptotic expansion of spectral heat content is stable under integrable perturbation when heat loss is sufficiently large. This is a joint work with Renming Song and Tomasz Grzywny. 10/9/2017 7:30:00 PM10/9/2017 8:30:00 PMFalse
Matthew Junge, Duke University Matthew Junge, Duke University Ewing 336 Title: The bullet problem with discrete speeds Abstract: A bullet is fired along the real line each second with independent uniformly random speeds from [0,1]. When two bullets collide they mutually annihilate. The still open bullet problem asks if the first bullet ever survives. We establish survival in the variant where speeds are discrete. Joint with Brittany Dygert, Christoph Kinzel, Annie Raymond, Erik Slivken, and Jennifer Zhu. 4/17/2017 5:30:00 PM4/17/2017 6:30:00 PMFalse
Subhro Ghosh, Princeton UniversitySubhro Ghosh, Princeton UniversityEwing 336Title: Large holes in particle systems : forbidden regions, large deviations and potential theory Abstract: In particles systems, a ``hole'' of size R is defined to be a ball of radius R that is devoid of particles. The study of how the probability of having such a hoe decays to 0 (as R -> infty) is an important and well-studied question in particle systems. In this talk, we ask what causes a large hole to appear ? In other words, conditioned on having a large hole, how does the configuration of particles outside the hole look like ? Surprisingly, very little is understood about this question, except in the very special case of Gaussian random matrices, where there is an accumulation of particles at the edge of the hole, and equilibrium intensity beyond. We study this question in the context of zeros of Gaussian random polynomials, and provide a complete description of the intensity profile of the outside particles. A remarkable feature that we find is the appearance of a curious ``forbidden region'' between the accumulation at the edge of the hole and the equilibrium intensity far beyond. This is in stark contrast to the case of Gaussian random matrices, and seems to be novel even in the wider setting of statistical physics models. Our methods connect to large deviation principles for random polynomials, potential theory and constrained optimization on measure spaces. These ideas can also be applied to other problems, including Jancovici-Lebowitz-Manificat laws for Coulomb systems at general temperatures, and understanding over and under-crowding phenomena for Gaussian zeros. Based on joint works with Alon Nishry. 4/10/2017 5:30:00 PM4/10/2017 6:30:00 PMFalse

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