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Abstract: In this talk, we will define fractional powers of nondivergence form elliptic operators in bounded domains under minimal regularity assumptions. We will characterize a Poisson problem driven by such operators with a degenerate/singular extension problem. We then develop the method of sliding paraboloids in the Monge–Ampère geometry to prove Harnack inequality for classical solutions to the extension equation. This in turn implies Harnack inequality for solutions to the fractional Poisson problem. This work is joint with Pablo Raúl Stinga (Iowa State University).
Link to paper: https://arxiv.org/abs/2012.14779
Speaker's Contact Information:
Email: maryv@utexas.edu
Website: https://maryvaughan.github.io/
Abstract: In the study of geometric flows it is often important to understand when a flow which converges along a sequence of times going to infinity will, in fact, converge along every such sequence of times to the same limit. While examples of finite dimensional gradient flows that asymptote to a circle of critical points show that this cannot hold in general, a positive result can be obtained in the presence of a so-called Lojasiewicz-Simon inequality. In this talk I will discuss some aspects of a recent joint work with Melanie Rupflin and Peter M. Topping in which we examined this problem for a geometric flow that is designed to evolve a map describing a closed surface in a given target manifold into a parametrization of a minimal surface. On one hand, we were able to construct explicit targets so that the flow exhibited non-uniqueness. On the other hand, when the target is real analytic, we were able to prove a Lojasiewicz-Simon inequality and show convergence to a unique limit in the absence of singular behaviour.
Link to paper: https://doi.org/10.1515/acv-2019-0086
Speaker's Contact Information:
Email: kohout@maths.ox.ac.uk
Abstract: The Mehta integral is the canonical partition function for 1-dimensional log-Coulomb gas in a harmonic potential well. Mehta and Dyson showed that it also determines the joint probability densities for the eigenvalues of Gaussian random matrix ensembles, and Bombieri later found its explicit form. We introduce the p-adic analogue of the Mehta integral as the canonical partition function for a p-adic log-Coulomb gas, discuss its underlying combinatorial structure, and find its explicit formula and domain.
Link to paper: https://arxiv.org/abs/2001.03892
Speaker's Contact Information:
Email: jwebster@uoregon.edu
Website: https://pages.uoregon.edu/jwebster/
Abstract: In 1995 Kim famously proved the Ramsey bound R(3,t) \ge c t^2/\log t by constructing an n-vertex graph that is triangle-free and has independence number at most C \sqrt{n \log n}. We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph K_n into a packing of such nearly optimal Ramsey R(3,t) graphs.
More precisely, for any \epsilon>0 we find an edge-disjoint collection (G_i)_i of n-vertex graphs G_i \subseteq K_n such that (a) each G_i is triangle-free and has independence number at most C_\epsilon \sqrt{n \log n}, and (b) the union of all the G_i contains at least (1-\epsilon)\binom{n}{2} edges. Our algorithmic proof proceeds by sequentially choosing the graphs G_i via a semi-random (i.e., Rodl nibble type) variation of the triangle-free process.
As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo (concerning a Ramsey-type parameter introduced by Burr, Erdos, Lovasz in 1976). Namely, denoting by s_r(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H=K_3 and establish that s_r(K_3) = \Theta(r^2 \log r).
Subject codes: 05C55, 05C80, 05D10, 60C05
Link to paper: https://link.springer.com/article/10.1007/s00493-019-3921-7
Speaker's Contact Information:
Email: he.guo@gatech.edu