Georgia Institute of Technology College of Sciences

High throughput genetic sequencing arrays with thousands of measurements per sample and a great amount of related censored clinical data have increased demanding need for better measurement specific model selection. In this paper we establish strong oracle properties of non-concave penalized methods for non-polynomial (NP) dimensional data with censoring in the framework of Cox's proportional hazards model. A class of folded-concave penalties are employed and both LASSO and SCAD are discussed specifically. We unveil the question under which dimensionality and correlation restrictions can an oracle estimator be constructed and grasped. It is demonstrated that non-concave penalties lead to significant reduction of the "irrepresentable condition" needed for LASSO model selection consistency. The large deviation result for martingales, bearing interests of its own, is developed for characterizing the strong oracle property. Moreover, the non-concave regularized estimator, is shown to achieve asymptotically the information bound of the oracle estimator. A coordinate-wise algorithm is developed for finding the grid of solution paths for penalized hazard regression problems, and its performance is evaluated on simulated and gene association study examples.

In this talk I will present recent work, in collaboration with J.Dolbeault, G. Tarantello and A. Tertikas,about the symmetry properties of extremal functions for (interpolation)functional inequalities playing an important rolein the study of long time behavior of evolution diffusion equations.Optimal constants are rarely known,in fact one can write them explicitely only when the extremals enjoymaximal symmetry. This is why the knowledge of the parameters' regionswhere symmetry is achieved is of big importance. In the case of symmetrybreaking, the underlying phenomena permitting it are analyzed.

The Seiberg-Witten equations, introduced by Edward Witten in 1994, are a first-order semilinear geometric PDE that have led to manyimportant developments in low-dimensional topology. In this talk,we study these equations on cylindrical 4-manifolds with boundary, which we supplement with (Lagrangian) boundary conditions that have a natural Morse-Floer theoretic interpretation. These boundary conditions, however, are nonlinear and nonlocal, and so the resulting PDE is highlyunusual and nontrivial. After motivating and describing the underlying geometry for the Seiberg-Witten equations with Lagrangian boundary conditions, we discuss some of the intricate analysis involved in establishing elliptic regularity for these equations, including tools from the pseudodifferential analysis ofelliptic boundary value problems and nonlinear functional analysis.

A well-known conjecture of Lovasz and Plummer asserts that the number of perfect matchings in 2-edge-connected cubic graph is exponential in the number of vertices. Voorhoeve has shown in 1979 that the conjecture holds for bipartite graphs, and Chudnovsky and Seymour have recently shown that it holds for planar graphs. In general case, however, the best known lower bound has been until now barely super-linear. In this talk we sketch a proof of the conjecture. The main non-elementary ingredient of the proof is Edmonds' perfect matching polytope theorem. This is joint work with Louis Esperet, Frantisek Kardos, Andrew King and Daniel Kral.