AMS - IMS- SIAM Joint Summer Research Conference
Monday, June 25 – Thursday, July 5
Mount Holyoke College
South Hadley, Massachusetts
William Beckner (co-chair), University of Texas - Austin
Harmonic analysis, broadly understood as the study of the decomposition of functions and operators into their basic constituents, is a mathematical subject with roots that go back hundreds of years. Despite its age, the subject continues to flourish. Its techniques and results are central to much of modern analysis, and the area is influenced by and has applications to a wide range of other mathematical topics. These include linear and nonlinear partial differential equations, differential and integral geometry, number theory, complex analysis, representation theory, and probability and mathematical physics. In the first half of the twentieth century harmonic analysis was closely linked to complex function theory and Lebesgue integration, but during the last fifty years more sophisticated real variable methods were developed which allowed applications to a variety of new problems. The activity in this field shows no sign of abating. During the last ten years there has been important progress on a number of outstanding problems. The object of this two-week conference is to provide an opportunity for both young and established researchers to exchange ideas and to consolidate recent progress. It is impossible to briefly summarize all of the topics that may be discussed at the conference, but the following very short list may give some sense of possible directions.
1. Oscillatory integrals and geometric measure theory-- Two outstanding questions in Fourier analysis are the Bochner-Riesz problem, which deals with the problem of convergence of Fourier integrals in several variables, and the restriction problem, which asks about the size of Fourier transforms on lower-dimensional sets. These questions are closely connected to questions about the behavior of oscillatory integrals depending on a parameter. Recent deep results in this area have partially depended on progress in problems in geometric measure theory concerning lower bounds for the dimension of Besicovich sets.
2. Classical singular integrals-- There have been a number of important recent results on questions related to the classical theory of singular integrals. These include results and applications of the boundedness of the bilinear Hilbert transform, applications of singular integral theory to situations in which classical "doubling conditions" are not satisfied, product-type singular integrals, and discrete versions of classical singular integrals and maximal functions with applications to ergodic theory.
3. Applications to partial differential equations-- Many methods in harmonic analysis were developed to solve partial problems in partial differential equations. As an example, recently a wealth of hard estimates on linear and multilinear oscillatory integrals was obtained in order to understand better the solutions of wave and Schrödinger-type equations and many of their nonlinear variants. Moreover, the use of singular integral methods has led to progress on the equations in fluid mechanics.
4. Estimates for Fourier integral operators-- In many variable coefficient situations estimates for various classes of Fourier integral operators are needed. In recent years some research has focused on the situation where the wavefront relation of the operator is not necessarily the graph of a canonical transformation. Although considerable progress has been made, many questions remain wide open.
5. Analysis on Lie groups-- Lie groups provide a natural setting for many questions in harmonic analysis. Moreover, the solution to a problem on a group can often be used as a model for solutions in more general situations. It is expected that some talks will address recent progress dealing with various questions on nilpotent Lie groups concerning local solvability of linear differential operators, singular integrals and applications to complex analysis. Moreover, recently there have been advances on the Kunze-Stein phenomenon on semisimple groups, on bounds for spherical functions, and on estimates for solutions of the heat and wave equations in various situations.
Because of the large number of topics, this conference will last for two weeks. Sessions will run Monday, June 25 through Friday, June 29, and Sunday, July 1 through Thursday, July 5. There will be no sessions on Saturday, June 30.
More information about transportation, lodging, etc. will be available at the AMS website .