Bordeaux

Consider all planar graphs on n vertices. What is the smallest graph that contains them all as induced subgraphs? We provide an explicit construction of such a graph on $n^{4/3+o(1)}$ vertices, which improves upon the previous best upper bound of $n^{2+o(1)}$, obtained in 2007 by Gavoille and Labourel.

In this talk, we will gently introduce the audience to the notion of so-called universal graphs (graphs containing all graphs of a given family as induced subgraphs), and devote some time to a key lemma in the proof. That lemma comes from a recent breakthrough by Dujmovic et al. regarding the structure of planar graphs, and has already many interesting consequences - we hope the audience will be able to derive more. This is based on joint work with Cyril Gavoille and Michal Pilipczuk.

The chromatic number of a graph is trivially bounded from above by the maximum degree plus one, and from below by the size of a largest clique. Reed proved in 1998 that compared to the trivial upper bound, we can always save a number of colors proportional to the gap between the maximum degree and the size of a largest clique. A key step in the proof deals with how to spare colors in a graph whose every vertex "sees few edges" in its neighborhood. We improve the existing approach, and discuss its applications to Reed's theorem and strong edge coloring.  This is joint work with Thomas Perrett (Technical University of Denmark) and Luke Postle (University of Waterloo).

We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f,\: t \geq 0$. The solutions $u = e^{tG_b}f$ with eigenfunctions $f$ of the generator $G_b$ with eigenvalues $\lambda,\: \Re \lambda

We present a new class of self interacting Markov chain models. In contrast to traditional Markov chains, their time evolution may depend on the occupation measure of the past values. We propose a theoretical basis based on measure valued processes and semigroup technics to analyze their asymptotic behaviour as the time parameter tends to infinity. We exhibit different types of decays to equilibrium depending on the level of interaction. In the end of the talk, we shall present a self interacting methodology to sample from a sequence of target probability measures of increasing complexity. We also analyze their fluctuations around the limiting target measures.