Michael Christ

"Illposedness of the nonlinear Schrodinger equation "

Wellposedness of the nonlinear Schrodinger equation in Sobolev spaces has been extensively studied, and it has been shown that wellposedness holds provided the Sobolev exponent exceeds a certain threshold, depending on the spatial dimension and the degree of nonlinearity.
In the reverse direction, relatively few isolated results are known indicating that these thresholds are optimal, especially for defocusing equations, for which blowup arguments do not apply and soliton-type solutions are unavailable. We show that the known thresholds are optimal in all cases. The key ingredient is a small dispersion analysis, in which the leading-order approximation to the PDE is a nonlinear ODE.
This is joint work with J. Colliander and T. Tao.

"The d-bar Neumann problem, magnetic Schrodinger operators, and the Aharonov-Bohm phenomenon"

The d-bar Neumann problem is a non-coercive boundary value problem for Laplace's equation on domains in C^n. The regularity properties of solutions depend on the complex geometry of the boundary, in a manner which has been extensively studied but is still only partially understood.
One fundamental problem is to characterize compactness of the Neumann operator. A sufficient condition, known as (P), has been introduced and studied by Catlin and Sibony. We investigate the necessity of condition (P) for the special class of domains possessing a one-dimensional symmetry group.
For these domains, both (P) and compactness are equivalent to certain properties of Schrodinger operators with both electric and magnetic fields. The relation between (P) and compactness is then closely linked with diagmagnetic and paramagnetic inequalities.
Our main result is an example showing how another effect, quite different from (P), can create compactness. The construction is based on an extreme form of the Aharonov-Bohm effect.
However, our example is not at all smooth, and we present strong evidence that for smoothly bounded domains with symmetry, property (P) is equivalent to compactness.
This is joint work with S. Fu.

Ronald Coifman

"Challenges in Analysis: High Dimensional Geometry and Approximation"

The so called curse of dimensionality is well known in statistics and other fields involving dependence on a large number of parameters. In these lectures we make the point that these are issues involving Harmonic Analysis. The talks will discuss effective functional and geometric approximation in high dimensions. In particular we will discuss various issues involved in approximating empirical functions of a large number of parameters including geometric analysis of data sets embedded in high dimensions. Such analysis can be achieved through Harmonic Analysis and operator theory on the data. We will also discuss effective low dimensional functional approximation (around 10-20 dimensions ). These mathematical issues will be illustrated on a variety of examples from biology, chemistry, mutimedia ..

Alex Iosevich

"Analysis and combinatorics of distances set"

Let E be a subset of the unit cube in dimensions two or greater. Let D_K(E) denote the set of distances between pairs of elements of E with respect to the distance induced by a convex body K symmetric with respect to the origin.The Falconer Distance Problem (FDP) asks whether one can conclude that D_K(E) has positive Lebesgue measure if the Hausdorff dimension of E is sufficiently large. Let S be a finite discrete subset of Euclidean space in dimensions two or greater. The Falconer Distance Problem can be viewed as a natural continuous analog of the Erdos Distance Problem (EDP) which asks for the smallest possible size of D_K(S) in terms of the size of S. We shall discuss the FDP and EDP, and their dependence on the geometric properties of K. We shall also discuss applications to problems in analysis and geometric combinatorics.

Gerd Mockenhaupt

"On the Hardy-Littlewood majorant property".

Hardy and Littlewood observed that Lp-spaces on the torus have the majorant property if p is a positive even integer. For other values of p it is known that the majorant property fails to hold. We will discuss a linearized variant of the majorant problem which relates it to restriction problems for Fourier series to frequency sets E contained in an finite interval [0,N]. One is then asking for bounds on the quantity Bp(E) in:

While for a random selection of a frequency set E contained in an interval of length N the constants Bp(E) are at most of logarithmic growth in N there are sets Ep in [0,N] for which one has power growth in N (provided p is not an even integer). This is joint work with Wilhelm Schlag.

Camil Muscalu

"Multilinear singular integrals, Part II".

Three years ago, Christoph Thiele gave a talk at the Riviere-Fabes Symposium, where he presented a theorem he obtained in collaboration with Terry Tao and myself, which generalized the results on the bilinear Hilbert transform.

The purpose of my talk is to describe what happened afterwords with this theory of multilinear singular integrals and their Carleson type maximal analogs. Most of the results are joint work with Terry Tao and Christoph Thiele ".

Mikhail Safonov

"Mean value theorem for harmonic functions: some unusual applications"

We discuss two applications of the classical mean value theorem (MVT) for harmonic (or subharmonic) functions. The first one is based only on a simple consequence of the MVT, so-called growth theorem, which is also true for more general second order elliptic equations Lu=0. We use it in order to control the boundary behavior of solutions to the equation Lu=f in a bounded domain, where f may blow near the boundary. As another application, we show that the well-known interior Schauder type estimates for solutions to the Poisson equation can be derived on the grounds of the MVT solely. These two facts are the core of the theory of intermediate Schauder estimates, which was developed by D. Gilbarg, L. Hörmander, and J. H. Mikhael. Their methods use a barrier technique, which works only for Lipschitz domains. By our approach, this theory can be extended to more general domains.