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Nonlinear and Linearized Models in Elasticity 4.12.2023 ? Lennart Machill ? Oberseminar Differentialgeometrie, Universität Münster Many variational models in elasticity are nonlinear and nonconvex, resulting in difficult numerical approximations and high computational costs. The derivation of simplified effective theories still preserving the main features of the original systems plays therefore a significant role in current research. Despite the long history of the subject, the theory flourished in the last twenty years triggered by the availability of the rigidity estimate in [Friesecke, James, Müller ?02]. We discuss the latter estimate and give an introduction to variational theories in elasticity. A particularly interesting case is the approximation of thin three-dimensional solids with lower dimensional models, which requires analysing such inequalities with respect to the domain thickness. In this context, we will include recent results in viscoelasticity and discuss a nonlinear Korn inequality.

Displays were introduced by Zink in the 2000s in order to generalize Dieudonné theory to arbitrary p-complete base rings. When studying Shmiura varieties and related objects one is naturally led to the task of defining group-theoretic versions of Abelian varieties and p-divisible groups. We report on results in the direction of developing a theory of (G, mu)-displays for a datum (G, mu), where G is a parahoric Zp-group scheme, extending work of Bültel-Pappas and Pappas. We furthermore give an application to the EKOR-stratification on the special fiber of a Kisin-Pappas integral Shimura variety.

Inspired by problems from fluid dynamics, we introduce an approach to obtain lower bounds for Lyapunov exponents of stochastic PDEs. Our proof relies on the introduction of a novel Lyapunov functional for the projective process associated to the equation, based on the study of dynamics of the energy median and on a notion of non-degeneracy of the noise that leads to high-frequency stochastic instability. Joint work with Martin Hairer.

We will explain some recent developements in algebraic K-Theory and highlight similarities (and differences) to the situation in operator K-Theory. The central object that we consider is the category of non-commutative motives a la Blumberg--Gepner--Tabuada. We argue that this should be seen as an algebraic analogue of Kasparov's KK-category. Through recent developments of Efimov and Claussen--Scholze we can now transport some methods from the KK-category to the category of non-commutative motives, specifically give similar models of assembly maps. If time permits we explain applications to geometric topology.

Kolloquium über Geschichte und Didaktik der Mathematik