In this seminar I will present a semi-langrangian discretisation of the Monge-Ampère operator, which is of interest in optimal transport 
and differential geometry as well as in related fields of application.

I will discuss the proof of convergence to viscosity solutions. To address the challenge of uniqueness and convexity we draw upon the classical relationship with Hamilton-Jacobi-Bellman equations, which we extend to the viscosity setting. I will explain that the monotonicity of semi-langrangian schemes implies that they possess large stencils, which in turn requires careful treatment of the boundary conditions.

The contents of the seminar is based on current work with X Feng from the University of Tennessee.

*/ /*-->*/ Let G be a locally compact Hausdorff topological group. Examples are Lie groups, p-adic groups, adelic groups, and discrete groups. The BC (Baum-Connes) conjecture proposes an answer to the problem of calculating the K-theory of the convolution C* algebra of G. Validity of the conjecture has implications in several different areas of mathematics --- e.g. Novikov conjecture, Gromov-Lawson-Rosenberg conjecture, Dirac exhaustion of the discrete series, Kadison-Kaplansky conjecture. An expander is a sequence  of finite graphs which is efficiently connected. Any discrete group which contains an expander as a sub-graph of its Cayley graph is a counter-example to  the BC conjecture with coefficients. Such discrete groups have been constructed by Gromov-Arjantseva-Delzant and by Damian Osajda. This talk will indicate how to make a correction in BC with coefficients. There are no known counter-examples to the corrected conjecture, and all previously known confirming examples remain confirming examples.

Can we extend the FEM to general polytopic, i.e. polygonal and polyhedral, meshes while retaining 
the ease of implementation and computational cost comparable to that of standard FEM? Within this talk, I present two approaches that achieve just  that (and much more): the Virtual Element Method (VEM) and an hp-version discontinuous Galerkin (dG) method.

The Virtual Element spaces are like the usual (polynomial) finite element spaces with the addition of suitable non-polynomial functions. This is far from being a novel idea. The novelty of the VEM approach is that it avoids expensive evaluations of the non-polynomial "virtual" functions by basing all 
computations solely on the method's carefully chosen degrees of freedom. This way we can easily deal 
with complicated element geometries and/or higher continuity requirements (like C1, C2, etc.), while 
maintaining the computational complexity comparable to that of standard finite element computations.

As you might expect, the choice and number of the degrees of freedom depends on such continuity 
requirements. If mesh flexibility is the goal, while one is ready to  give up on global regularity, other approaches can be considered. For instance, dG approaches are naturally suited to deal with polytopic meshes. Here I present an extension of the classical Interior Penalty dG method which achieves optimal rates of convergence on polytopic meshes even under elemental edge/face degeneration. 

The next step is to exploit mesh flexibility in the efficient resolution of problems characterised by 
complicated geometries and solution features, for instance within the framework of automatic FEM 
adaptivity. I shall finally introduce ongoing work in this direction.

Erdos and Renyi introduced a model for studying random graphs of a given "density" and proved that there is a sharp threshold at which lower density random graphs are disconnected and higher density ones are connected.  Motivated by ideas in geometric group theory we will explain some new threshold theorems we have discovered for random graphs.  We will then, explain applications of these results to the geometry of Coxeter groups.  Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.