Suppose we replace the nonlinearity in the KdV equation by a more general polynomial nonlinearity. What can we now say about local well-posedness? When the polynomial contains a quadratic term of the form \(uu_{xx}\), a Mizohata-type integrability condition acts as an obstruction to well-posedness in \(H^s\). Several authors have studied the problem using weighted H^s spaces, with the disadvantage that these are no longer translation invariant. In this talk we will investigate local well-posedness in the translation invariant \(l^1H^s\) spaces of Marzuola, Metcalfe and Tataru.

Littlewood–Paley theory is a simple technique that has proven invaluable in harmonic analysis and PDE. In many cases, it offers a concrete and conceptually straightforward replacement for the somewhat obscure machinery of interpolation theory. In this talk I will give a gentle introduction to the basic ideas and outline a few applications. The goal will be to prove a "fractional Lorentz rule".
[ Notes for the talk ]

We consider an initial value problem for a quadratically nonlinear inviscid Burgers–Hilbert equation that models the motion of vorticity discontinuities. We use a normal form transformation, which is implemented by means of a near-identity coordinate change of the independent spatial variable, to prove the existence of small, smooth solutions over cubically nonlinear time-scales. For vorticity discontinuities, this result means that there is a cubically nonlinear time-scale before the onset of filamentation.

Rearrangement inequalites are a simple but powerful technique in analysis. We will introduce the notion of the symmetric decreasing rearrangment of a function and discuss Riesz's rearrangement inequality as well as the generalization introduced by Brascamp, Lieb and Luttinger. If time permits we will discuss applications to finding extremizers and sharp constants.

Using estimates on random Fourier series I will explain how to construct (a version of) the Gibbs measure for the NLS on the torus. I will rely on some very classical harmonic analysis and results of Leibowitz, Rose, Speer and Bourgain.

Exactly 100 years ago Weyl published the density law for eigenvalues of the Laplacian on a bounded domain. He solved conjectures of Lorentz and Hilbert, motivated by earlier work of Lord Rayleigh and by the study of black body radiation. Eigenvalues can be interpreted as energy levels and Weyl laws and their refinements are known in many settings.

When systems are open or have some damping, quantum states, in addition to «rest energy», also have «decay rates» and the behaviour of their joint densities is richer and more mysterious. It was first studied by Regge 50 years ago.

One thing that emerges for classically chaotic systems is a fractal Weyl law, with a scaling law given by the dimension of the classical repeller. This was first suggested by Sjöstrand and has led to investigations in mathematics and physics.

I will present the statements in some simple settings and will illustrate them with numerical examples.

The \(k\)-plane transform is the operator that maps a function and a \(k\)-plane to the integral of the function over the \(k\)-plane. It satisfies an \(L^p\to L^q\) dilation-invariant inequality. I will explain how its symmetry properties lead to the value of the best constant, solving the endpoint case of a conjecture from Baernstein and Loss. This essentially uses the theory of competing symmetries developed by Carlen and Loss, and is an application of the rearrangement inequalities that we have heard about earlier in this seminar.

If time permits, I will discuss the question of the uniqueness of extremizers.

Notes for the talk: arXiv:1111.5061

The \(U^{p}\) and \(V^{p}\) spaces have been heavily used in the work of Herr, Hadac, Koch, and Tataru. These spaces are atomic spaces whose atoms are linear solutions to a dispersive partial differential equations. For critical problems for long time they provide an important refinement of the \(X^{s, b}\) spaces.

Kadomtsev–Petviashvili (KP) equation is a natural generalization of Korteweg–de Vries (KdV) equation in 2 dimension. It models shallow water waves in \(x\) direction, with some mild dispersion in \(y\)-direction. In this talk, I will review results about Initial Value Problems for KP-I and KP-II equations, and in particular discuss the paper by Ionescu, Kenig and Tataru, where they proved global wellposedness for KP-I in the energy space.

Exterior Differential Systems is a tool which allows one to phrase a partial differential equation in geometric language. In addition to giving a coordinate free expression to the PDE, this allows many of the tools from Geometry to be used to attack these problems. In special situations, such as those arising from geometric problems we can use symmetries of the problem to make calculation easier. In this talk I will introduce some of the basic tools and methods of EDS. As an application of this I will show how a Lie algebra arises (at least locally) from a Lie group.

I will present Pour-El and Richards' 1981 example of computable initial conditions for the wave equation such that the resulting unique solution is not computable. Central to the construction is the fact that the solution operator for the wave equation is unbounded (in the uniform norm). Similar results can be obtained for any closed unbounded linear operator (subject to some mild computability conditions). No prior knowledge of recursion theory will be assumed.

We discuss extremizers for a family of Fourier restriction inequalities on planar curves whose curvature satisfies a natural geometric assumption. It turns out that any extremizing sequence of nonnegative functions has a subsequence which converges to an extremizer. We hope to describe the method of proof, which is of concentration compactness flavor, in some detail. Tools include bilinear estimates, a variational calculation and a modification of the usual method of stationary phase.