Vorträge in der Woche 03.07.2017 bis 09.07.2017

Montag, 03.07.2017: Lie algebras and Jordan algebras

Prof. Dr. Ernest Vinberg (Moscow)

Lie algebras and Jordan algebras are two important classes of non-associative algebras. Lie algebras are widely known, because they appear in the theory of Lie groups. Jordan algebras are mainly interesting due to their relations to Lie algebras. To each Jordan algebra J one can associate three Lie algebras embedded into one another: the algebra of inner derivations of J, the structure algebra of J, and the so-called Tits--Kantor--Koecher construction, which is, roughly.speaking, the algebra of traceless matrices of order 2 over J. In particular, the simple Lie algebras of types F_4, E_6, E_7 are associated to the exceptional simple 27-dimensional Jordan algebra (the Albert algebra). The talk will be devoted to yet another construction, which associate a Lie algebra to each semisimple Jordan algebra of (generalized) degree 3. This Lie algebra contains the Tits--Kantor--Koecher construction as a proper subalgebra. In this way one can uniformly obtain all the simple Lie algebras but A_n and C_n, together with some natural embeddings between them.

Dienstag, 04.07.2017: Cuspidal types and Characters II

Steven Charlton

Mittwoch, 05.07.2017: Numerical Analysis of the Evolving Surface Finite Element Method for some Parabolic Problems

Christian Andreas Power (Universität Tübingen)

Numerische Analyse von partiellen Differentialgleichungen auf bewegten Oberflächen stellen ein sehr aktives Forschungsgebiet dar. Ich habe mich auf parabolische Probleme spezialisiert und werde die vier Paper, die ich während meiner Dissertation erarbeitet habe, vorstellen und die zentralen Ergebnisse erläutern. Es wird vorgestellt: - Eine Arbeit über die Konvergenz von Arbitrary-Eulerian-Lagrangian finite Elemente mit Zeitdiskretisierung höherer Ordnung für ein parabolisches Problem, - eine Arbeit über die Konvergenz von finiten Elementen mit Zeitdiskretisierung höherer Ordnung für ein quasilineares Problem, - eine Arbeit über die Konvergenz von finiten Elementen für ein parabolisches Problem in der Maximumsnorm, - eine Arbeit über die Konvergenz von finiten Elementen für ein System, wobei die Bewegung der Oberfläche gekoppelt ist an die Lösung einer parabolischen Gleichung auf derselben Oberfläche

Donnerstag, 06.07.2017: Shape Differentation: New Perspectives

Prof. Dr. Ralf Hiptmair (ETH Zürich)

R. Hiptmair1, J.-Z. Li 2 A. Paganini3, S. Sargheini1 1 Seminar for Applied Mathematics, ETH Zürich, Switzerland 2 SUSTC, Shenzhen, PR China 3 Mathematical Institute, University of Oxford, UK The presentation examines the "derivative" of solutions of second-order boundary value problems and of output functionals based on them with respect to the shape of the domain. A rigorous approach relies on encoding shape variation by means of deformation vector fields, which will supply the directions for taking shape derivatives. These derivatives and methods to compute them numerically are key tools for studying shape sensitivity, performing gradient based shape optimization, and small-variation shape uncertainty quantification. A unifying view of second-order elliptic boundary value problems recasts them in the language of differential forms (exterior calculus). Fittingly, the shape deformation through vector fields matches the concept of Lie derivative in exterior calculus. This paves the way for a unified treatment of shape differentiation in the framework of exterior calculus. The obtained formulas can be employed in the so-called adjoint approach to derive shape gradients of concrete output functionals. The resulting expressions allow different reformulations. Though equivalent for exact solutions of the involved boundary value problems, they deliver vastly different accuracies in the context of finite element approximation, as confirmed by a rigorous asymptotic a priori convergence analysis for a number of important cases.

Donnerstag, 06.07.2017: Deformation of metrics towards constant scalar curvature

Prof. Dr. Simon Brendle (Columbia University)

The classical uniformization theorem asserts that any Riemannian metric on a closed two- dimensional surface is conformal to a metric of constant Gaussian curvature. A higher dimensio- nal analogue of this statement is given by the solution of the Yamabe problem: Any metric on an n-dimensional manifold is conformal to a metric of constant scalar curvature. This problem is equivalent to the existence of a positive solution of the nonlinear elliptic equation of the form $\Delta u - \frac{n-2}{4(n-1)} R u + c u^{\frac{n+2}{n-2}} = 0$. In this lectures, I will describe the background of this problem, and its variational formulation in terms of the Yamabe functional. The gradient flow associated with the Yamabe functional leads to an curvature flow, and I will discuss why this flow converges to a metric of constant scalar curvature for any initial metric.