Georgia Institute of Technology College of Sciences

The KLS conjecture says that the Cheeger constant of any logconcave density is achieved to within a universal, dimension-independent constant factor by a hyperplane-induced subset. Here we survey the origin and consequences of the conjecture (in geometry, probability, information theory and algorithms) and present recent progress resulting in the current best bound, as well as a tight bound for the log-Sobolev constant (both with Yin Tat Lee). The conjecture has led to several techniques of general interest.

An overarching problem in matrix weighted theory is the so-called A2 conjecture, namely the question of whether the norm of a Calderón-Zygmund operator acting on a matrix weighted L2 space depends linearly on the A2 characteristic of the weight. In this talk, I will discuss the history of this problem and provide a survey of recent results with an emphasis on the challenges that arise within the setup.

The restricted three body problem models the motion of a body of zero mass under the influence of the Newtonian gravitational force caused by two other bodies, the primaries, which describe Keplerian orbits. In 1922, Chazy conjectured that this model had oscillatory motions, that is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. Its existence was not proven until 1960 by Sitnikov in a extremely symmetric and carefully chosen configuration. In 1973, Moser related oscillatory motions to the existence of chaotic orbits given by a horseshoe and thus associated to certain transversal homoclinic points. Since then, there has been many atempts to generalize their result to more general settings in the restricted three body problem.In 1980, J. Llibre and C. Sim\'o, using Moser ideas, proved the existence of oscillatory motions for the restricted planar circular three body problem provided that the ratio between the masses of the two primaries was arbitrarily small. In this talk I will explain how to generalize their result to any value of the mass ratio. I will also explain how to generalize the result to the restricted planar elliptic three body problem. This is based on joint works with P. Martin, T. M. Seara. and L. Sabbagh.

We present and analyze a novel sparse polynomial approximation method for the solution of PDEs with stochastic and parametric inputs. Our approach treats the parameterized problem as a problem of joint-sparse signal reconstruction, i.e., the simultaneous reconstruction of a set of signals sharing a common sparsity pattern from a countable, possibly infinite, set of measurements. Combined with the standard measurement scheme developed for compressed sensing-based polynomial approximation, this approach allows for global approximations of the solution over both physical and parametric domains. In addition, we are able to show that, with minimal sample complexity, error estimates comparable to the best s-term approximation, in energy norms, are achievable, while requiring only a priori bounds on polynomial truncation error. We perform extensive numerical experiments on several high-dimensional parameterized elliptic PDE models to demonstrate the superior recovery properties of the proposed approach.