Events for 05/04/2022 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Nabile Boussaid, Université Franche--Comte, Besançon

Title: Inverse Regge poles problem on a warped ball

Abstract: We study a new type of inverse problem on warped product Riemannian manifolds with connected boundary that we name warped balls. Using the symmetry of the geometry, we first define the set of Regge poles as the poles of the meromorphic continuation of the Dirichlet-to-Neumann map with respect to the complex angular momentum appearing in the separation of variables procedure. These Regge poles can also be viewed as the set of eigenvalues and resonances of a one-dimensional Schrödinger equation on the half-line, obtained after separation of variables. Secondly, we find a precise asymptotic localisation of the Regge poles in the complex plane and prove that they uniquely determine the warping function of the warped balls. Joint work with Jack Borthwick and Thierry Daudé.

Numerical Analysis Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Dr J.M. Melenk, TU Wien

Title: Stability and convergence of Galerkin discretizations of the Helmholtz equation in piecewise smooth media

Abstract: We consider the Helmholtz equation with variable coefficients at large wavenumber k. In order to understand how k affects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit in k. At the heart of our analysis is the decomposition of solutions into two components: the first component is a piecewise analytic, but highly oscillatory function and the second one has infinite regularity but features wavenumber-independent bounds. This decomposition generalizes earlier decompositions of Melenk and Sauter (2010 & 2022) which considered the Helmholtz equation with constant coefficients, to the case of piecewise analytic coefficients. This regularity theory for the Helmholtz equation with variable coefficients allows for the analysis of high order Galerkin discretizations of the Helmholtz equation that are explicit in the wavenumber k. We show that quasi-optimality is guaranteed under the following scale resolution condition: (a) the approximation order p is selected as p = O(log k) and (b) the mesh size h is such that kh/p is sufficiently small. This scale resolution condition ensures quasi-optimality for a variety of time-harmonic wave propagation problems including FEM-BEM coupling and Maxwell problems.

Groups and Dynamics Seminar

Time: 3:00PM - 4:00PM

Location: online

Speaker: Danny Calegari, U Chicago

Title: Sausages and Butcher Paper

Abstract: For each q>1 the Shift Locus of degree q is the space of monic depressed degree q polynomials in one complex variable for which every critical point is in the attracting basin of infinity. We give an explicit description of the Shift Locus as a (combinatorial) building whose pieces turn out to be homeomorphic to affine complex algebraic varieties.