Colin Jahal: Some progress on the unique ergodicity problem
Thursday, 20.05.2021 10:30 im Raum via Zoom
In 2005, Kechris, Pestov and Todorcevic exhibited a correspondence between combinatorial properties of structures and dynamical properties of their automorphism groups. In 2012, Angel, Kechris and Lyons used this correspondence to show the unique ergodicity of all the minimal actions of some subgroups of $S_\infty$. In this talk, I will give an overview of the aforementioned results and discuss recent work generalizing results of Angel, Kechris and Lyons in several directions.
Many geometric applications of PDEs depend on transversality results, e.g. in gauge theory or symplectic topology. One needs certain moduli spaces of solutions to elliptic PDEs such as Seiberg-Witten monopoles or pseudoholomorphic curves to be "transversely cut out", so that they are smooth and have the expected dimension. A problem arises: these PDEs often come with natural symmetries, and everyone knows that you cannot have transversality and symmetry at the same time. In this talk, I will outline a general strategy to deal with transversality without sacrificing symmetry, and discuss what conditions need to be checked before you can apply it to your favorite PDE.
In this seminar I will review the theory of flat G-chains, as they were introduced by H. W. Fleming in 1966, and currents with coefficients in groups. One of the most recent development of the theory concerns its application to the Steiner tree problem and other minimal network problems which are related with a Eulerian formulation of the branched optimal transport. Starting from a 2016 paper by A. Marchese and myself, I will show how these problems are equivalent to a mass-minimization problem in the framework of currents with coefficients in a (suitably chosen) normed group.
A Jordan curve is a simple loop on the sphere. We recently introduced the conformally invariant Loewner energy to measure the roundness of a Jordan curve. Initially, the definition is motivated by describing asymptotic behaviors of Schramm-Loewner evolution (SLE), a probabilistic model of random curves of importance in statistical mechanics. Intriguingly, this energy is shown to be finite if and only if the curve is a Weil-Petersson quasicircle, an interesting class of Jordan curves that has more than 20 equivalent definitions arising in very different contexts, including Teichmueller theory, geometric function theory, hyperbolic geometry, and string theory. The myriad of perspectives on this class of curves studied since the eighties is both luxurious and mysterious. In my talk, I will overview the basics of Loewner energy, SLE, and Weil-Petersson quasicircles and show you how ideas from probability theory inspire many new results on Weil-Petersson quasicircles.
Abstract: Localized spot patterns, where one or more solution components concentrates at discrete points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. A theoretical study of such patterns in a 3-D setting is, largely, a new frontier. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) and Schnakenberg systems in the limit of a small activator diffusivity. Depending on the range of parameters, spot patterns can undergo competition instabilities, leading to spot-annihilation events, or shape-deforming instabilities triggering spot self-replication. In the absence of these instabilities, spots evolve slowly according to an ODE gradient flow, given by a discrete energy defined in terms of the reduced-wave Green's function. Open problems for localization on higher co-dimension structures are discussed.