Departement Mathematik

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Zurich colloquium in mathematics

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Frühjahrssemester 2025

Datum / Zeit Referent:in Titel Ort
11. März 2025
16:30-18:15 		Charles Bordenave
Institute de Mathématiques de Marseille
Details

Zurich Colloquium in Mathematics

Titel Random perturbation of Toeplitz matrices
Referent:in, Affiliation Charles Bordenave, Institute de Mathématiques de Marseille
Datum, Zeit 11. März 2025, 16:30-18:15
Ort KO2 F 150
Abstract Toeplitz matrices form a rich and ubiquitous class of possibly non-normal matrices. Their asymptotic spectral analysis in high dimension is well-understood, as illustrated by the strong Szegö limit theorem for Toeplitz determinants. The spectra of these matrices are notoriously highly sensitive to small perturbations. In this talk, we explore the spectrum of a banded Toeplitz matrix perturbed by a random matrix in the asymptotic of high dimension. We show that the outlier eigenvalues are driven by a low-dimensional random analytic matrix field alongside an explicit deterministic matrix that captures the algebraic structure of the resonances responsible for the outlier eigenvalues. Along the way, we present new variations around the strong Szegö limit theorem. The talk is based on a joint work with Mireille Capitaine and François Chapon.
Random perturbation of Toeplitz matricesread_more
KO2 F 150
29. April 2025
16:30-17:15 		Endre Suli
University of Oxford
Details

Zurich Colloquium in Mathematics

Titel Hilbert's 19th problem and discrete De Giorgi{Nash{Moser theory: analysis and applications
Referent:in, Affiliation Endre Suli, University of Oxford
Datum, Zeit 29. April 2025, 16:30-17:15
Ort KO2 F 150
Abstract Models of non-Newtonian fluids play an important role in science and engineering and their mathematical analysis and numerical approximation have been active fields of research over the past decade. This lecture is concerned with the analysis of numerical methods for the approximate solution of a system of nonlinear partial differential equations that arise in models of chemically-reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra-filtrate of blood plasma that contains hyaluronic acid, whose function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where the power-law exponent depends on a spatially varying nonnegative concentration function, expressing the concentration of hyaluronic acid, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations one has to derive a uniform Hölder norm bound on the sequence of approximations to the concentration in a setting where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely a bounded function with no additional regularity. This necessitates the development of a discrete counterpart of the De Giorgi--Nash--Moser theory, which is then used, in conjunction with various compactness techniques, to prove the convergence of the sequence of numerical approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration
Hilbert's 19th problem and discrete De Giorgi{Nash{Moser theory: analysis and applicationsread_more
KO2 F 150

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