The study of singular solutions of the NLS goes back to the 1960s, with applications in nonlinear optics and, more recently, in BEC. Asymptotic and numerical studies conducted in the 80s showed that singular solutions of the critical NLS collapse with the Townes (R) profile at a blowup rate known as the loglog law. Recently (2003) Merle and Raphael proved this result rigorously for a large class of initial conditions. Concurrently, it was demonstrated experimentally that the profile of collapsing laser beams is given by the Townes profile. Thus, all the research that was carried out from the eighties until these days leads to the belief that the Townes profile is the only attractor of blowup solutions of the critical NLS.
In this talk I will present new families of singular solutions of the critical and supercritical NLS that collapse with a self-similar ring profile, and whose blowup rate is different from the one of the "old" singular solutions. I will show, experimentally and theoretically, that these new blowup profiles are attractors for large super-Gaussian initial conditions.
In addition, I will present in the talk a semi-static adaptive grid method we have developed for the numerical simulations involved in this study for solutions which focus over 15 orders of magnitude.

We classify compact ancient solutions of the curve
shortening flow and the Ricci flow on Surfaces. We show that these are either a family of contracting circles (contracting spheres in the case of the Ricci flow on surfaces), which is a type I ancient solution,or a family of Angenent ovals (Rosenau solutions in the case of the Ricci flow on surfaces), which corresponds to a type II
solution.

The question of global regularity for the 3D Navier-Stokes equations is a major open problem in applied analysis. It is well-known that the existence and uniqueness of strong solutions could be obtain under suitable additional assumptions. In this talk I will review some old and new results about the sufficient conditions for the global regularity to the 3D Navier-Stokes equations.

In this talk, I will present a large margin method for hierarchical
classification. The main focus here is to utilize the dependency
structure among classes to improve the classification performance of
flat classification. In such a situation, flat classification is
infeasible in the presence of a large number of dependent classes, which
occurs often in gene function discovery. Various hierarchical losses
will be discussed, in addition to an application to
gene function prediction.