If $G$ is an irreducible lattice in a semisimple Lie group, every action of $G$ on a tree has a global fixed point. I will give an elementary discussion of Y. Shalom's proof of this result, focussing on the case of $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$. Emphasis will be placed on the geometric aspects of the proof and on the importance of reduced cohomology, while other representation theoretic/functional analytic tools will be relegated to a couple of black boxes.

This talk will hopefully highlight the general framework in which Penrose tilings are proved to be aperiodic and in fact a tiling. 

In the game 'Set', players compete to pick out groups of three cards sharing common attributes. But how many cards must be dealt before such a group must appear? 
This is an example of a "cap set problem", a problem in Ramsey theory: how big can a set of objects get before some form of order appears? We will translate the cap set problem into a problem of geometry over finite fields, discussing the current best upper bounds and running through an elementary proof. We will also (very) briefly discuss one or two implications of the cap set problem over F_3 to other questions in Ramsey theory and computational complexity
 

We provide the rate of convergence of general monotone numerical schemes for parabolic Hamilton-Jacobi-Bellman equations in bounded domains with Dirichlet boundary conditions. The so-called "shaking coefficients" technique introduced by Krylov is used. This technique is based on a perturbation of the dynamics followed by a regularization step by convolution. When restricting the equation to a domain, the perturbed problem may not satisfy such a restriction, so that a special treatment near the boundary is necessary.