Manifolds with corners are similar to manifolds, yet are locally modelled on subsets $[0,\infty)^k \times R^{n-k}$. I will discuss some of the theory of these objects, as well as introducing $C^\infty$-rings. This will explain the background to my current research in $C^\infty$-Algebraic Geometry. Time permitting, I will briefly discuss my current research on $C^\infty$-schemes with corners and motivation of this research.

Hyperbolic groups were introduced by Gromov and generalize the fundamental groups of closed hyperbolic manifolds. Since a closed hyperbolic manifold is aspherical, it is a classifying space for its fundamental group, and a hyperbolic group will also admit a compact classifying space in the torsion-free case. After an introduction to this and other topological finiteness properties of hyperbolic groups and their subgroups, we will meet a construction of R. Kropholler, building on work of Brady and Lodha. The construction gives an infinite family of hyperbolic groups with finitely-presented subgroups which are non-hyperbolic by virtue of their finiteness properties. We conclude with progress towards determining minimal examples of the "sizeable" graphs which are needed as input to the construction.

Consider a family of uniformly bounded $W^{2,p}$ isometric immersions of an $n$-dimensional (semi-) Riemannian manifold into (resp., semi-) Euclidean spaces. Are the weak limits still isometric immersions?

We answer the question in the affirmative for $p>n$ in the Riemannian case, by exploiting the div-curl structure of the Gauss-Codazzi-Ricci equations, which describe the curvature flatness of the isometric immersions. Along the way a generalised div-curl lemma in Banach spaces is established. Moreover, the endpoint case $p=n=2$ is settled. 

In the semi-Riemannian case we reduce the problem to the weak continuity of H. Cartan's structural equations in $W^{1,p}_{\rm loc}$, which is proved by a generalised compensated compactness theorem relating the weak continuity of quadratic forms to the principal symbols of differential constraints. Again for $p>n$ we obtain the weak rigidity. The case of degenerate hypersurfaces are also discussed, as well as connections to PDEs in fluid dynamics.

We report on recent developments in the study of nonlocal operators. The central object of the talk are quadratic forms similar to those that define Sobolev spaces of fractional order. These objects are naturally linked to Markov processes via the theory of Dirichlet forms. We provide regularity results for solutions to corresponding integrodifferential equations. Our emphasis is on forms with singularand anisotropic measures. Some of the objects under consideration are related to the Boltzmann equation, which leads to an interesting question of comparability of quadrativ forms. The talk is based on recent results joint with B. Dyda and with K.-U. Bux and T. Schulze.

We prove tight upper and lower bounds on an observable of the antiferromagnetic Potts model. From this we deduce the case d=3 of a conjecture of Galvin and Tetali on maximising the number of proper colourings in d-regular graphs.