Georgia Institute of Technology College of Sciences

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

In this talk I will describe a theory of matrix completion for the extreme case of noisy 1-bit observations. In this setting, instead of observing a subset of the real-valued entries of a matrix M, we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M. The central question I will address is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but I will show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M when $\|M\|_{\infty} \le \alpha$ and $\rank(M) \le r$. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this maximum likelihood estimate by optimizing a convex program. I will also provide lower bounds showing that this estimate is near-optimal and illustrate the potential of this method with some preliminary numerical simulations.

We consider a so-called "One sided bump conjecture", which gives asufficient condition for two weight boundedness of a Calderon-Zygmundoperator. The proof will essentially use the Corona decomposition, which isa main tool for a first proof of $A_2$ (also, $A_p$ and $A_p-A_\infty$)conjecture. We will focus on main difficulty, that does not allow to get afull proof of our one sided bump conjecture.

I will describe a joint work with Vincent Millot (Paris 7) where we investigate the singular limit of a fractional GL equation towards the so-called boundary harmonic maps.