The microscopic structure of a macroscopic system in a steady state is described locally, i.e. at a suitably scaled macroscopic point $x$, by a time invariant measure of the corresponding infinite system with translation invariant dynamics. This measure may be extremal, with good decay of correlations, or a superposition of extremal measures, with weights depending on $x$ (and possibly even on the way one scales).

I will illustrate the above by some exact results for 1D lattice systems with two types of particles (plus holes) evolving according to variants of the simple asymmetric exclusion process, in open or closed systems. Somewhat surprisingly, the spatially asymmetric local dynamics satisfy (in some cases) detailed balance with respect to a global Gibbs measure with long range pair interactions.

Let S be a fixed surface, and let k and q be fixed integers. Is there a polynomial-time algorithm that decides whether an input graph of girth at least q drawn in S is k-colorable? This question has been studied extensively during the last 15 years. We will briefly survey known results.

Then we will describe a solution to one of the two cases left open (the prospects for the other one are not bright). For every surface S we give a polynomial-time algorithm that computes the chromatic number of an input triangle-free graph G drawn in S. The new contribution here is deciding whether G is 3-colorable, and has two main ingredients. The first is a coloring extension theorem that makes use of disjoint paths in order to construct a coloring. The notion of "winding number" of a 3-coloring plays an important role. The second ingredient is a theorem bounding the number and sizes of faces of size at least four in 4-critical triangle-free graphs on a fixed surface, a generalization of a theorem of Thomassen.

By developing more structure theory and using the notion of bounded expansion of Nesetril and Ossona de Mendez we were able to implement the algorithm to run in linear time. This is joint work with Zdenek Dvorak and Daniel Kral.

The main motivation for the theory of mock modular forms comes from the desire to provide a framework in which we can understand the mysterious and intriguing mock theta functions, as well as related functions, like Appell functions and theta functions associated to indefinite quadratic forms.

In this talk, we will describe the nature of the modularity of the original mock theta functions, formulate a general definition of mock modular forms, and describe further examples. We will also consider a generalization to higher depth mock modular forms

Erdős and Posa proved that a there exists a function $f$ such that any graph either has $k$ disjoint cycles or there exists a set of $f(k)$ vertices that intersects every cycle. The analogous statement is not true for odd cycles - there exist numerous examples of graphs that do not have two disjoint odd cycles, and yet no bounded number of vertices intersects every odd cycle. However, Reed has given a partial characterization of when there does not exist a bounded size set of vertices intersecting every odd cycle.

We will discuss recent work on when a graph has many disjoint cycles of non-zero length modulo $m$ for arbitrary $m$. When $m$ is odd, we see that again there exists a function $f$ such that any graph either has $k$ disjoint cycles of non-zero length modulo $m$ or there exists a set of at most $f(k)$ vertices intersecting every such cycle of non-zero length. When $m$ is even, no such function $f(k)$ exists. However, the partial characterization of Reed can be extended to describe when a graph has neither many disjoint cycles of non-zero length modulo $m$ nor a small set of vertices intersecting every such cycle.

In a couple of recent works with C. Khare and M. Larsen we contruct finte groups of Lie type B_n, C_n and G_2 as Galois groups over rational numbers. The method combines some established, special cases of the functoriality principle with l-adic representations attached to self-dual automrophic representations of GL(n).