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IAS/Park City Mathematics Institute
Graduate Summer School 2003

Harmonic Analysis and Partial Differential Equations

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. Candidates should have completed basic graduate courses, including introductory courses in real and functional analysis.  Some background at the advanced undergraduate level in complex analysis, Fourier analysis, and partial differential equations is recommended. In general, these students will have completed their second year and in many cases will already be working on a thesis. While a majority of the participants will be graduate students, some postdoctoral scholars and researchers may also be interested in attending.

The main activity of the Graduate Summer School will be a set of intensive short courses offered by the leaders in the field; these lectures will not duplicate standard courses available elsewhere. Each course will consist of lectures with problem sessions. Course assistants will be available for each lecture series.

The Graduate Summer School bridges the gap between a general graduate education in mathematics and the specific preparation necessary to do research on problems of current interest. The main activity of the Graduate Summer School is a set of intensive short courses offered by world leaders in their fields. These lectures, which do not duplicate course material available elsewhere, are designed to introduce students to exciting current research in mathematics.  The participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Course Titles and Descriptions:

Elias Stein

Title: Singular integrals and several complex variables

Abstract: We will discuss two stages in the application of singular integrals to problems of several complex variables. Our focus will be the (relative) solving operator for the Kohn-Laplacian $ \square_b$ on $ q$-forms, and the corresponding Cauchy-Szegö projection on the null space of $ \bar\partial_b$.

We first study the situation where the Calderón-Zygmund paradigm applies: where the boundary of the domain is of finite type and the eigenvalues of the Levi-form are comparable. We then take up a class of model domains: those which are ``decoupled.'' These give good illustrations of how the ideas of the product theory, flag kernels, lifting, and projections can be applied.

 

Guy David

Title:  Uniform rectifiability and applications

Abstract: Uniformly rectifiable sets are closed subsets $ E$ of $ {{\mathbb{R}}}^n$ that are Ahlfors-regular (i.e., there is a measure $ \mu$ on $ E$ such that $ C^{-1}r^d \leq \mu (E \cap B(x,r)) \leq C r^d$ for $ x \in E$ and $ 0<r< diam(E)$, where $ d$ is a fixed integer) and satisfy some quantitative rectifiability condition. When $ d=1$, the condition is that $ E$ be contained in an Ahlfors-regular curve (connected set), for instance. We will give a few characterizations and basic properties of uniformly rectifiable sets, in particular of dimension $ d=1$, and describe situations where the notion is useful. We will discuss $ L^2$-boundedness of singular integrals on $ E$, regularity properties of minimizers and quasiminimizers for functionals with a surface term (Almgren restricted sets, minimizers of the Mumford-Shah functional), and connections to Menger curvature and analytic capacity.

 

Christoph Thiele

Title: Scattering Theory and Non-Linear Fourier Analysis

Abstract: Non-linear Fourier analysis plays a role in diverse areas such as scattering theory, spectral theory of Schrödinger operators, orthogonal polynomials, integrable systems, operator theory on spaces of analytic functions, Riemann-Hilbert factorization problems, interpolation problems, control theory. We plan to give a unified introduction into the area from a harmonic analysis point of view. Some focus will be on scattering theory of square integrable potentials on the line.

 

Gigliola Staffilani,

Title: Global well-posedness for dispersive equations and the method of almost conservation laws

Abstract: In these lectures we illustrate a new method to extend local well-posedness results for dispersive equations to global ones. The main ingredient of this method is the definition of a family of what we call almost conservation laws. In particular we analyze the Korteweg-de Vries initial value problem and we illustrate in general terms how the ``algorithm'' that we use to formally generate almost conservation laws can be used to recover the infinitely many conserved integrals that make the KdV an integrable system.

 

Carlos E. Kenig

Title:  Harmonic Analysis and its applications to non-linear evolution equations

Abstract: The course will deal with the application of pseudo-differential operators to establishing well-posedness (i.e.existence, uniqueness and continuous dependence) for quasi-linear Schrodinger equations. We will start out by reviewing the basic facts about classical pseudo-differential operators, and their application to the local smoothing effect for solutions to Schrodinger equations. We will then apply them to establish variants of the energy method, for Schrodinger equations, first in the semilinear case, (i.e. when the second order terms are linear), and finally extend the arguments to the quasi-linear case,where the second order terms are linear in the second derivatives, but have coefficients which depend non-linearly on the lower order derivatives.

 

Terence Chi-Shen Tao

Title: Kakeya-type problems, restriction conjectures, and local smoothing estimates

Abstract: We discuss the restriction problem - when can the Fourier transform of an L^p function be meaningfully restricted to a sphere or similar surface? This problem is connected to other problems in harmonic analysis, such as the Bochner-Riesz problem for summing Fourier series, and is also connected with the quantitative behaviour of solutions to linear equations such as the wave equation and the Schrodinger equation, in particular the problem of bilinear estimates and of the local smoothing conjecture. Interestingly, all the recent progress on these problems has come by combining Fourier-analytic methods (e.g. orthogonality, uncertainty principle) with geometric combinatorics results, and in particular recent progress on the Kakeya conjecture. We discuss all these interconnections, and applications, in this lecture series.

 

Participants in the Graduate Summer School also may wish to become involved in the Undergraduate Program, attend parts of the Research Program, or participate in the programs of the Education component. Graduate students are expected to participate in Institute-wide activities such as the "Cross Program Activities" and may be asked to contribute some time to volunteer projects related to running the Summer Session.

A limited number of graduate students who have not completed the basic courses may attend. These students will attend some graduate level courses and may be involved as teaching assistants in other programs or work as audio-visual assistants.