Date: Monday, February 05, 2018
Location: 3096 East Hall (4:00 PM to 5:00 PM)
Title: Polynomials in dynamics and the logarithmic energy problem on the real line with external fields
Abstract: The fundamental electrostatics problem concerns the equilbrium distribution of a unit charge on a conductor. If the conductor is regarded as a compact set $E$ in the complex plane and charges repel each other according to an inverse distance law, then in the absence of an external field, equilibrium will be reached when the total energy $\int \int log (zt)^{1}d\mu(z)d\mu(t)$ is minimal among all possible charge distributions (measures) $\mu$ on $E$ having unit charge 1. In this talk I will discuss the electrostatics problem with external field for closed sets in the plane, existence and descriptions of the support of the minimizer of the weighted energy problem and show how this problem relates to the problem of the description of zeroes sets of polynomials, to various notions of dynamics and to capacity of sets such as the filled Julia set. I will also discuss if time permits connections of the weighted energy problem to the best packing problem, the problem of uniformly distributing many points on a sphere and zero distribution of orthogonal and extremal polynomials. This is joint work with David Benko (Szeged), Peter Grabner (Graz), David Ragozin (Washington), Peter Dragnev (IPFW) and Arno Kuijlaars (Leuven).
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Speaker: Steven Damelin
Institution: Math Reviews
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