In the first half of this talk I will give an overview of the necessary semiclassical analysis to derive the semiclassical trace formula. I will then use these results along with the isoperimetric inequality to prove that the spectrum of an $n$-dimensional semiclassical radial Schr\"odinger operator determines the potential within a large class of potentials for which we assume no symmetry or analyticity.

Over the last few decades, a vast literature has developed concerning the mapping properties, between Lp spaces, of linear operators which involve singularities and/or curvature. Questions about extremizers for the associated inequalities have received less attention. Natural issues include the existence of extremizers, their identification when possible, uniqueness modulo symmetries, and quantitative and qualitative properties in cases when identification is not possible. I will discuss this last question, for one of the most canonical linear operator of this type. Extremizers satisfy a certain nonlinear Euler Lagrange equation. The main result is that all solutions of this equation are smooth. The main tool is a new family of weighted Lp inequalities. (joint work with Qingying Xue)

Subtle properties of harmonic measure were established in the 80's. These properties were concerned with the metric properties (mostly with the Hausdorff dimension) of measure. Works of Bourgain, Brennan, Carleson, Jones, Makarov, Wolff clarified the picture in 2D. Except for one "easy" problem when dim w < dim supp w? We will show that this problem can be reduced to a free boundary problem in certain interesting situations. In one case we will solve this free boundary problem.

In this talk we study the recent result of Killip, Stovall, and Visan concerning the nonlinear Klein-Gordon initial value problem with initial condition in the energy space. Combining the concentration compactness method with recent results proving global well-posedness and scattering of the mass critical Schrödinger initial value problem Killip, Stovall, and Visan were able to prove nonlinear Klein-Gordon equation is globally well-posed and scattering for all data in the defocusing case and data below a threshold in the focusing case.

In this talk a proof of the continuity of the norm of a general Calderon-Zygmund operator in a weighted L^p space as a function of the A_{p} norm of the weight is presented. The Riesz-Thorin interpolation theorem plays an important role in the proof. The sharpness of the estimate will also be discussed.

In this talk, I will review the regularity problem for Korteweg-de Vries (KdV) equation on the line, give a brief summary of the current well-posedness and ill-posedness results, and then discuss a possible way to get a-priori bounds and weak solution below the critical threshold H^-3/4.

Let u be a solution to the equation \Box_g u = 0 where g is some (nonstationary) Lorentzian metric and \Box_g its associated d'Alembertian. If we assume a priori that certain local energy norms for u and its higher derivatives hold, we can prove that u decays pointwise like t^{-3+} away from the light cone. As an application, we can prove the aforementioned decay on Kerr spacetimes and some perturbations. This is joint work with Jason Metcalfe and Daniel Tataru.

In 1964, Hiroshi Fujita and Tosio Kato proved that the Navier-Stokes equations were locally well-posed in time for initial data in the homogeneous Sobolev space $\dot{H}^{\frac{1}{2}}$. Let us call $S$ the subset of functions in $\dot{H}^{\frac{1}{2}}$ which generate global solutions. In a recent work, Gallagher, Planchon and Iftimie have shown that $S$ is unbounded, arc-connected and open. I will recall the proof of the Fujita-Kato theorem and talk about these properties.

Many classical topics in the harmonic analysis of Euclidean space have analogues in symmetric spaces. I will try to outline something of this theory of symmetric spaces, paying particular attention to understanding the Radon transform, spherical functions and (time permitting) Poisson's equation on rank one symmetric spaces.

We discuss an alternative model to the well known Black Scholes model and deduce an interesting limit that arises. The financial interpretation is described in the title, and mathematically this turns out to be a problem related to asymptotic analysis of parabolic PDE with gradient constraints.

We examine a certain class of trilinear integral operators which incorporate oscillatory factors $e^{iP}$, where $P$ is a real-valued polynomial, and prove smallness of such integrals in the presence of rapid oscillations. Tools include sublevel set estimates, higher dimensional versions of van der Corput's lemma and corresponding multilinear analogues. This is joint work with Michael Christ.

Suppose that a compactly supported measure on Rd has the restriction condition, that is, there is an adjoint Fourier restriction inequality associated to it. We are interested in proving existence of extremizers for that inequality. Here we show that in the non endpoint case there are extremizers and we provide examples where extremizers fail to exist in the endpoint case. This is from Fanelli, Vega and Visciglia : "On the existence of maximizers for a family of restriction Theorems".

In 1970s, Hirota developed the direct method (so called bilinear method), a powerful way to obtain multisoliton solutions to many nonlinear evolution equations such as KdV and KP equations. Based upon Hirota's discovery, Sato introduced the KP hierarchy stressing that KP equation should be thought of as belonging to an infinite family of mutually compatible equations. Shortly after Sato's work, Date-Jimbo-Kashiwara-Miwa found an explanation to the KP hierarchy from a representation theoretic point of view. I will try to outline how Kac-Moody Lie algebras enter into this picture.