Events for 09/28/2022 from all calendars

Noncommutative Geometry Seminar

Time: 2:00PM - 3:00PM

Location: BLOC 302

Speaker: Rudolf Zeidler, University of Munster

Title: Nonnegative scalar curvature on manifolds with at least two ends

Abstract: I will present an obstruction to positive scalar curvature (psc) on complete manifolds with at least two ends based on the existence of incompressible hypersurfaces that do not admit psc. This result mixes an analytic technique based on $\mu$-bubbles, an augmentation of the classical minimal hypersurface obstructions to psc, with a topological argument based on positive scalar curvature surgery. Due to the latter a surprising (but necessary!) spin condition enters our result even though our methods are not based on the Dirac operator. Concretely, let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and $Y\subset M$ a two-sided closed connected incompressible hypersurface that does not admit a metric of psc. Suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Then $M$ does not admit a complete metric of psc. As a consequence, our result answers questions of Rosenberg-Stolz and Gromov up to dimension $7$. Joint work with Simone Cecchini and Daniel Räde.

URL: Event link

Numerical Analysis Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Loic Cappanera, University of Houston

Title: Robust numerical methods for incompressible flows with variable density

Abstract: The modeling and approximation of incompressible flows with variable density are important for a large range of applications in biology, engineering, geophysics and magnetohydrodynamics. Our main goal here is to develop and analyze robust numerical methods that can be used with high order finite element and spectral methods. We first discuss the main challenges we face before introducing a semi-implicit scheme based on projection methods and the use of the momentum, equal to the density times the velocity, as primary unknown. We present an analysis of the stability and convergence properties of the method and obtain a priori error estimates. A fully explicit version of the scheme is then proposed. Its robustness and convergence are studied with a pseudo spectral code over various setups involving large ratio of density, gravity and surface tension effects, or manufactured solutions. Applications to magnetohydrodynamics instabilities in industrial setups such as aluminum production cells, and liquid metal batteries will be presented. Eventually, a novel method based on artificial compressibility techniques is introduced and its performances are compared to our projection-based method.