The classical Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts, to which the classical theory cannot apply. Particular examples include polynomial rings over non-semisimple algebras, extension algebras of modules, etc. In this talk I'll introduce a generalized Koszul theory which does not demand the semisimple property. It preserves many classical results as Koszul duality and has a close relation to the classical one. Applications of this generalized theory to extension algebras of modules and modular skew group algebras will be described.

Logarithmic Sobolev inequalities (LSIs) show up in several areas of analysis; in particular, in probability. In this talk I will give some background and applications of LSIs. I will also discuss some recent work and show how LSIs can be used to give a new proof of the classical result that the empirical law of eigenvalues of a sequence of random matrices converges weakly to its mean in probability.

In non-commutative Iwasawa theory K-theoretic properties of Iwasawa algebras, i.e. completed group rings of e.g. compact p-adic Lie groups play a crucial role. Such groups arise naturally as Galois groups attached to p-adic representations as for example on the Tate module of abelian varieties. In this talk we address in particular the question for which such groups the invariant $SK_1$ vanishes. We reduce this vanishing to a linear algebra problem for Lie algebras over arbitrary rings, which we solve for Chevalley orders in split reductive Lie algebras. Also we shall try to indicate what arithmetic consequences the vanishing of $SK_1$ has.

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.

Motivated by the question whether (some) Diophantine equations are related to special values of $\zeta$- or $L$-functions we first describe the origin of classical Iwasawa theory. Then we give a survey on generalizations of these ideas to non-commutative Iwasawa theory, a topic which has been developed in recent years by several mathematicians, including the author.