The interface between water and vacuum (governed by the "water wave equation"), and the interface between oil and water in sand (governed by the "Muskat equation") can develop singularities in finite time. Joint work with A. Castro, D. Cordoba, F. Gancedo, J. Gomez and M. Lopez.

In 1964 Arnold constructed an example of instabilities for nearly integrable systems and conjectured that generically this phenomenon takes place.
There has been big progress attacking this conjecture in the past decade. Jointly with Ke Zhang we present a new approach to this problem. It is based on a construction of crumpled and flower Normally Hyperbolic Invariant Cylinders. Once existence of these cylinders is shown to construct diffusion we apply Mather variational mechanism. A part of the project is also joint with P. Bernard.

We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law. While proving this appears to be very difficult, some approximations to it can be proved by developing number theoretic and analytic methods. The work report on is joint with A. Ghosh and A. Reznikov.

We explain in this talk how Ramanujan graphs can be used to devise optimal cycle codes and review how other graph families related to a construction proposed by Margulis yield interesting families of quantum codes with logarithmic minimum distance. We finish the talk by providing another simple graph theoretic construction with improved minimum distance which grows proportionally to the square root of the quantum code length. (This is joint work with Gilles Zemor.)

Both of these talks will be useful preparation for Helmut Hofer's up coming mini-course on polyfold theory on April 4th and 5th.

Both of these talks will be useful preparation for Helmut Hofer's up coming mini-course on polyfold theory on April 4th and 5th

In 1964 Arnold constructed an example of instabilities for nearly integrable systems and conjectured that generically this phenomenon takes place.
There has been big progress attacking this conjecture in the past decade. Jointly with Ke Zhang we present a new approach to this problem. It is based on a construction of crumpled and flower Normally Hyperbolic Invariant Cylinders. Once existence of these cylinders is shown to construct diffusion we apply Mather variational mechanism. A part of the project is also joint with P. Bernard.