In the early nineties Scheffer produced a complicated example of a nontrivial weak solution to the incompressible Euler equations, having compact support in space and time. Subsequent papers by Shnirelman produced other examples of quite irregular solutions by different, yet complicated, methods.

In a recent joint work with L\'aszl\'o Sz\'ekelyhidi we have used a suitable ``$h$--principle'' to produce solutions with the same behavior in a relatively simple way. Our approach answers to further questions left open by the works of Scheffer and Shnirelman and might be relevant in understanding a long--standing conjecture of Onsager. The same kind of analysis has relevant applications also to the theory of hyperbolic systems of conservation laws and shares some surprising similarities with aspects of the theory of fully developed turbulence.

***Please note that there will be an additional talk by the speaker at IAS

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.

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.