A cohomology theory is a way of associating a complex of abelian groups to a topological space. Study of such complexes gives geometric information about the structure of the space. Moreover if we have an action of a suitable group on a space we can associate another such complex to that space which carries more information. I will sketch main ideas of this construction and provide some examples.

A few years ago we have proved in collaboration with L. Flaminio that some non-trivial limit distributions for the horocycle flow must have compact support. In this talk we will refine that result and describe an existence/non-existence result for limiting distributions. In fact it turns out that whether limiting distributions exist or not depends on the geometry of the surface (via the eigenvalues of the Laplace operator) and on the observable under consideration. The main new idea is to express the precise results on the asymptotics of ergodic averages for the horocycle flow in terms of a dynamically defined cocycle which has the correct scaling property under the dynamics of the geodesic flow. Such cocycles are closely related to the invariant distributions of Flaminio-Forni and are analogous to the coycles constructed by A. Bufetov for asymptotic foliations of a Markov compactum (and in particular for area-preserving flows on higher genus surfaces).

This is joint work with A. Bufetov.

In combinatorial geometry one frequently wants to select a point or a set of points that meets many simplices of a given family. The two examples are choosing a point in many simplices spanned by points of some P in R^d, and choosing a small set of points which meets the convex hull of every large subset of P (the weak epsilon-net problem). I will present a new class of constructions that yield the first nontrivial lower bound on the weak epsilon-net problem, and improve the best bounds for several other selection problems. Joint work with Jiří Matoušek and Gabriel Nivasch.

We will discuss some arithmetic counting problems, ranging from the antique (how many squarefree integers are there in [0..N]?) to the au courant (conjectures of Bhargava and Cohen-Lenstra about the distributions of discriminants and of class groups.) When considered over function fields, these conjectures reveal themselves as having to do with stabilization of cohomology of moduli spaces of covers of curves, or Hurwitz spaces. We will report on progress on the topological study of Hurwitz spaces, which leads to information about arithmetic counting problems over function fields over finite fields; for instance, a version of Cohen-Lenstra "correct up to the constant" for F_q(t). If time permits I will try to give a picture of the rather general ensemble of arithmetic counting conjectures suggested by the method (e.g. -- for how many squarefree integers in [0..N] is there a totally real quintic extension of Q with discriminant N?) and explain how to prove versions of these conjectures in the much easier regime where "q goes to infinity first."

(joint work with Akshay Venkatesh and Craig Westerland)

Boltzmann's equation is known to converge, under a certain hydrodynamic regime, to an incompressible Navier-Stokes-Fourier system. It is only recently that the final steps to a mathematically rigorous and complete justification of this hydrodynamic convergence were provided. However, only certain types of intermolecular interactions, still physically unsatisfying, were considered.

We establish this hydrodynamic limit for the physically relevant case of long range intermolecular interactions. In this situation, the difficulty comes from the fact that the Boltzmann collision operator exhibits a rather complex nature due to a non-integrable singularity in the collision kernel.

If a closed subset X of the plane is projected orthogonally onto a line, then the Hausdorff dimension of the image is no larger than the dimension of X (since the projection is Lipschitz), and also no larger than 1 (since it is a subset of a line). A classical theorem of Marstrand says that for any such X, the projection onto almost every line has the maximal possible dimension given these constraints, i.e. is equal to min(1,dim(X)). In general, there can be uncountably many exceptional directions.

An old conjecture of Furstenberg is that if A, B are subsets of [0,1] invariant respectively under x2 and x3 mod 1, then for their product, X=AxB, the only exceptional directions in Marstrand's theorem are the two trivial ones, namely the projections onto the x and y axes. Recently, Y. Peres and P. Shmerkin proved that this is true for certain self-similar fractals, such as regular Cantor sets. I will discuss the proof of the general case, which relies on a method for computing dimension using local entropy estimates. I will also describe some other applications. This is joint work with Pablo Shmerkin.

Hadwiger's conjecture is a well known open problem in graph theory. It states that every graph with chromatic number k, contains a certain structure, called a "clique minor" of size k. An interesting special case of the conjecture, that is still wide open, is when the graph G does not contain three pairwise non-adjacent vertices. In this case, it should be true that G contains a clique minor of size t where t >= |V(G)|/2. This remains open, but Jonah Blasiak proved it in the subcase when |V(G)| is even and the vertex set of G is the union of three cliques. Here we prove a strengthening of Blasiak's result: that the conjecture holds if some clique in G contains at least |V(G)|/4 vertices.

This is a consequence of a result about packing ``seagulls''. A seagull in G is an induced three-vertex path. It is not known in general how to decide in polynomial time whether a graph contains k pairwise disjoint seagulls; but we answer this for graphs with no stable sets of size three.

This is joint work with Paul Seymour.

We will explain how toroidal compactifications of certain Kuga families of abelian varieties over integral models of PEL-type Shimura varieties, including for example all those products of universal abelian schemes, can be constructed by a uniform method. We will also explain some of their applications to the cohomology theories of automorphic bundles.

To simulate supernova explosions, one must solve simultaneously the non-linear, coupled partial differential equations of radiation hydrodynamics. What's more, due to a variety of instabilities and asymmetries, this must eventually be accomplished in 3D. The current state-of-the-art is 2D, plus rotation and magnetic fields (assuming axisymmetry). Nevertheless, with the current suite of codes, we have been able to explore the evolution of the high-density, high-temperature, high-speed environment at the core of a massive star at death. It is in this core that the supernova explosion is launched. However, the complexity of the problem has to date obscured the essential physics and mechanisms of the phenomenon, making it indeed one of the "Grand Challenges" of 21st century astrophysics. Requiring forefront numerical algorithms and massive computational resources, the resolution of this puzzle awaits the advent of peta- and exa-scale architectures and the software to efficiently use them. In this talk, I will review the current state of the science and simulations as we plan for the fully 3D, multi-physics capabilities that promise credibly to crack open this obdurate astrophysical nut.

We associate p-adic Galois representations to globally generic cusp forms on GSp(4), over a totally real field, with a Steinberg component at some finite place. At places v not dividing p one has local-global compatibility, the local correspondence being that defined by Gan and Takeda. In particular, the rank of the monodromy operator at such a place v is determined by the level of the v-component of the cusp form. Moreover, the Swan conductor is essentially the depth.