In recent years, ideas from the world of partial differential equations (PDEs) have found their way into the arena of graph and network problems. In this talk I will discuss how techniques based on nonlinear PDE models, such as the Allen-Cahn equation and the Merriman-Bence-Osher threshold dynamics scheme can be used to (approximately) detect particular structures in graphs, such as densely connected subgraphs (clustering and classification, minimum cuts) and bipartite subgraphs (maximum cuts). Such techniques not only often lead to fast algorithms that can be applied to large networks, but also pose interesting theoretical questions about the relationships between the graph models and their continuum counterparts, and about connections between the different graph models.

I will compare features of (classical) cohomology theory of groups to the rather exotic features of bounded (or continuous bounded) cohomology of groups.
Besides giving concrete examples I will state classical cohomological tools/features and see how (if) they survive in the case of bounded cohomology. Such will include the Mayer-Vietoris sequence, the transfer map, resolutions, classifying spaces, the universal coefficient theorem, the cup product, vanishing results, cohomological dimension and relation to extensions. 
Finally I will discuss their connection to each other via the comparison map.

I will talk about connections between the compressible and incompressible Navier-Stokes systems. In case of the compressible model, as the bulk (volume) viscosity is very high, the divergence of the velocity becomes small, in the limit it is zero and we arrive at the case of incompressible system. An important role here is played by the inhomogeneous version of the classical Navier-Stokes equations. I plan to discuss analytical obstacle appearing within the analysis. The considerations are done in the framework of regular solutions in Besov and Sobolev spaces. The results which will be discussed are joint with Raphael Danchin from Paris.

In this talk, we will discuss sequences of immersions from 2-discs into Euclidean with finite total curvature where the Willmore energy converges to zero (a minimal surface). We will show that away from finitely many concentration points of the total curvature, the surface converges strongly in $W^{2,2}$.  Furthermore, we have an energy identity for the total curvature, at the concentration points after a blow-up procedure we show that there is a bubble tree consisting of complete non-compact (branched) minimal surfaces of finite total curvature which are quantised in multiples of 4\pi. We will also apply this method to the mean curvature flow, showing that sequences of surfaces that are locally converging to a self-shrinker in L^2 also develop a bubble tree of complete non-compact (branched) minimal surfaces in Euclidean space with finite total curvature quantised in multiples of 4\pi. 

Joint work with Jen Pestana.