Events for 11/16/2022 from all calendars

Numerical Analysis Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 306

Speaker: Shawn Walker, Louisiana State University

Title: Curvature and the HHJ Method

Abstract: This talk shows how the classic Hellan--Herrmann--Johnson (HHJ) method can be extended to surfaces, as well as approximate curvature. We start by showing how HHJ can be extended to surfaces embedded in ℝ³ to solve the surface version of the Kirchhoff plate equation. The surface Hessian of the "displacement" variable is discretized by an HHJ finite element function. Convergence is established for all possible combinations of mixed boundary conditions, e.g. clamped, simply-supported, free, and the "4th" condition. Numerical examples are shown, some which use "point" boundary conditions as well as solving the surface biharmonic equation.

We also show how the surface HHJ method can be used to post-process a discrete Lagrange function on a given surface triangulation to yield an approximation of its surface Hessian, i.e. a kind of Hessian recovery. Moreover, we demonstrate that this scheme can be used to give convergent approximations of the *full shape operator* of the underlying surface using only the known discrete surface, even for piecewise linear triangulations. Several numerical examples are given on non-trivial surfaces that demonstrate the scheme.

Groups and Dynamics Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 506a

Speaker: Chris Shriver, University of Texas, Austin

Title: Non-equilibrium Gibbs states on a tree

Abstract: We consider two notions of statistical equilibrium for a probability-measure-preserving shift system: an “equilibrium state” maximizes a functional called the pressure while a “Gibbs state” satisfies a local equilibrium condition. Classical results of Dobrushin, Lanford, and Ruelle show that these notions are equivalent for Z^d systems, under some assumptions on the interaction, and the equivalence has been extended to arbitrary amenable groups. Barbieri and Meyerovitch have recently shown that one direction still holds for sofic groups: equilibrium states are always Gibbs. We will show that the converse fails in a nontrivial way using the example of the free boundary Ising state on an infinite regular tree (i.e. a free group): we show that for all temperatures below the uniqueness threshold this state is nonequilibrium over some sofic approximation, and below the reconstruction threshold it is nonequilibrium over every sofic approximation.