Georgia Institute of Technology College of Sciences

Modern imaging data are often composed of several geometrically distinct constituents. For instance, neurobiological images could consist of a superposition of spines (pointlike objects) and dendrites (curvelike objects) of a neuron. A neurobiologist might then seek to extract both components to analyze their structure separately for the study of Alzheimer specific characteristics. However, this task seems impossible, since there are two unknowns for every datum. Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms. Utilizing the methodology of Compressed Sensing, the geometric separation problem can indeed be solved both numerically and theoretically. For the separation of point- and curvelike objects, we choose a deliberately overcomplete representation system made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures). The decomposition principle is to minimize the $\ell_1$ norm of the representation coefficients. Our theoretical results, which are based on microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved. This project was done in collaboration with David Donoho (Stanford University) and Wang-Q Lim (TU Berlin).

In this talk, we will discuss recent results by Ozawa on whether every compact submanifold of S^3 can be unknotted by twistings.

We will start explaining and formulating the mathematical questions involved in justifying statistical physics from dynamical first principles. We will particularly discuss the approach, suggested by Boltzmann, based on deriving effective equations for the distribution function of a particle system. This will lead us to Boltzmann kinetic equation and its H-principle. This corresponds to Chapters 1 and 2 of Dorfman "An introduction to Chaos in Non-equilibrium Statistical Mechanics".

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)