Georgia Institute of Technology College of Sciences

A discussion of the paper "Modeling and automation of sequencing-based characterization of RNA structure" by Aviran et al (PNAS, 2011).

The relative isoperimetric inequality inside an open, convex cone C states that under a volume constraint, the ball intersected the cone minimizes the perimeter inside C. In this talk, we will show how one can use optimal transport theory to obtain this inequality, and we will prove a corresponding sharp stability result. This is joint work with Alessio Figalli.

Mark Kac proposed in 1956 a program for deriving the spatially homogeneous Boltzmann equation from a many-particle jump collision process. The goal was to justify in this context the molecular chaos, as well as the H-theorem on the relaxation to equilibrium. We give answers to several questions of Kac concerning the connexion between dissipativity of the many-particle process and the limit equation; we prove relaxation rates independent of the number of particles as well as the propagation of entropic chaos. This crucially relies on a new method for obtaining quantitative uniform in time estimates of propagation of chaos. This is a joint work with S. Mischler.

This dissertation investigates the statistical learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning. Given a random couple (X,Y)\in R^d\times {\pm 1} with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. The prediction rule is constructed based on the observations (X_i,Y_i)_{i=1}^n sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for performance of the active learning methods under certain assumptions. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem. The second part of this work is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P, with X taking its values in some metric space S and Y - in a bounded subset of R. Given a collection of functions H={h_t}_{t\in \mb T} mapping S to R, the goal of dictionary learning is to construct a prediction rule for Y given by a linear (or convex) combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance. Finally, we provide similar bounds in the density estimation framework.

In this introductory talk, we review the dynamical motivation for definingsymplectic manifolds, then describe a class of invariants called symplecticcapacities, which are closely related to both volume and the existence ofperiodic orbits. We explore the connections and differences between thesethree notions in the context of some basic phenomena/problems in symplecticgeometry: Gromov's nonsqueezing theorem, the difference between symplecticand volume-preserving diffeomorphisms, and the question of existence ofclosed characteristics on energy surfaces.

In the first part of the talk we will give a brief survey of significant results going from S. Brown pioneering work showing the existence of invariant subspaces for subnormal operators (1978) to Ambrozie-Muller breakthrough asserting the same conclusion for the adjoint of a polynomially bounded operator (on any Banach space) whose spectrum contains the unit circle (2003). The second part will try to give some insight of the different techniques involved in this series of results, culminating with a brilliant use of Carleson interpolation theory for the last one. In the last part of the talk we will discuss additional open questions which might be investigated by these techniques.