Harmonic Analysis

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. In general, these students will have completed their first year of graduate school, and in some cases, may 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 leaders in the field, designed to introduce students to exciting, current research in mathematics. These short courses 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 participants of the Graduate Summer School meet three times each day for lectures, with one or two problem sessions scheduled each day as well.

Harmonic analysis is a central field of mathematics with a number of applications to geometry, partial differential equations, probability, and number theory, as well as physics, biology, and engineering. The Graduate Summer School will feature mini-courses in geometric measure theory, homogenization, localization, free boundary problems, and partial differential equations as they apply to questions in or draw techniques from harmonic analysis. The goal of the program is to bring together students and researchers at all levels interested in these areas to share exciting recent developments in these subjects, stimulate further interactions, and inspire the new generation to pursue research in harmonic analysis and its applications.

Student preparation: We seek motivated students interested in the mathematical aspects of harmonic analysis and related fields. Though familiarity with some of the topics listed above would be helpful, the formal prerequisites are a graduate-level course in real analysis as well as a comfort with advanced linear algebra. Additional prerequisites for individual courses are listed below with the course descriptions and include: geometric measure theory, partial differential equations, and probability theory.

The 28th Annual PCMI Summer Session will be held July 1 - 21, 2018.

2018 GSS Lecture Schedule: COMING SOON!!

2018 Organizers

Carlos Kenig, University of Chicago
Fang-Hua Lin, Courant Institute
Svitlana Mayboroda, University of Minnesota
Tatiana Toro, University of Washington

Week 1: July 1–7
Week 2: July 8–14
Week 3: July 15–21

Guy David, Université de Paris-Sud

Existence and regularity for minimal sets of dimension 2

There was a lot of recent activity on topics related to the Plateau problem, where one tries to find a surface of minimal area under some boundary constraints, such as being bounded by a given curve. In this course we will probably concentrate on one way to state boundary constraints, based on a notion of deformations of a set that preserves the boundary.

The objects in this theory are sets, as in earlier work of F. Almgren and J. Taylor, so we will not need so much structure. We will concentrate on $2$-dimensional sets, and nice boundaries that are either curves or surfaces, because this is a situation where we can hope for a reasonably nice description of minimal set.

This starts with the description of minimal cones with some specific boundary constraints, and then we can try to establish local regularity, i.e., H\"older or $C^1$ local equivalence to some simple models. We expect that such regularity results will help prove existence results for some types of Plateau problems. We may try to describe a systematic way to do this.

The main prerequisite is a knowledge of measure theory. Rectifiability will be used, but it is not needed to know this in advance. We may need to admit some (easy to believe but long to prove) earlier results.

Camillo De Lellis, Universität Zürich

Center manifolds

In my course I will analyze a toy case, that of a single sheet, where one canprove direct $C^{3,\alpha}$ regularity for a minimal surface without using Schauder's estimates and with elementary considerations, following an early paper by Spadaro and myself (Center manifold: a case study). If time allows I will give some ideas  about the center manifold needed in Almgren's theory and its youngest relatives.

Prerequisites: basic knowledge of elliptic partial differential equations (harmonic functions and their properties), of harmonic analysis (maximal functions and related estimates) and elementary geometric measure theory (Hausdorff measures, rectifiable sets, area formula) are enough to understand the toy case in full details.

Steve Hofmann, University of Missouri

An Introduction to Harmonic Analysis on Non-Smooth Sets

This course will be an introduction to harmonic analysis on non-smooth sets, that is, Lipschitz graphs and beyond. The material serves as a starting point for the theory of elliptic equations in domains with rough boundaries. A tentative list of topics to be covered is as follows:

• T1/Tb theory;
• Singular integrals on Lipschitz surfaces (Coifman-McIntosh-Meyer Theorem);
• Square functions;
• Extension of singular integral and square function theory beyond Lipschitz graphs.

Prerequisites. Theory of measure and integration. Ideally, some familiarity with the basics of harmonic analysis in Euclidean space (e.g., Hardy-Littlewood maximal function, basic Calderón-Zygmund theory).

Svetlana Jitomirskaya, University of California, Irvine

Arithmetic spectral transitions: a competition between hyperbolicity and the arithmetics of small denominators

We will describe a method to prove 1D Anderson localization in the regime of positive Lyapunov exponents that has allowed to solve the arithmetic spectral transition (from absolutely continuous to singular continuous to pure point spectrum) problem for the almost Mathieu operator, in coupling, frequency and phase. On the other end of the arithmetic spectrum, we will also present a sharp arithmetic criterion for the transition to full spectral dimensionality in the singular continuous regime, for the entire class of analytic quasiperiodic potentials.

Lectures will start with general background on discrete ergodic Schrodinger operators, as well as on cocycles, the Lyapunov exponents, Osceledec theorem, Furstenberg Theorem. We will then present the framework of the method and illustrate it with the simplest application: to the iidrv Anderson model. Then we will move to the quasiperiodic case and the interplay between the arithmetics and hyperbolicity.

A very captivating question in solid state physics is to determine/understand the hierarchical structure of spectral features of operators describing 2D Bloch electrons in perpendicular magnetic fields, as related to the continued fraction expansion of the magnetic flux. In particular, the hierarchical behavior of the eigenfunctions of the almost Mathieu operators, despite significant numerical studies and even a discovery of Bethe Ansatz solutions has remained an important open challenge even at the physics level.

We will sketch how the presented method leads to a complete solution of this problem in the exponential sense throughout the entire localization regime. Namely, we will describe, with very high precision, the continued fraction driven hierarchy of local maxima, and a universal (also continued fraction expansion dependent) function that determines local behavior of all eigenfunctions around each maximum, thus giving a complete and precise description of the hierarchical structure. In the regime of Diophantine frequencies and phase resonances there is another universal function that governs the behavior around the local maxima, and a reflective-hierarchical structure of those, a phenomena not even described in the physics literature. Lectures are based on a recent work joint with W. Liu and another work joint with S. Zhang.

Prerequisites: Spectral theory of bounded self-adjoint operators, in particular, the spectral measures; basic ergodic theory (ideally, up to Kingman's subadditive ergodic theorem), basic continued fractions. Also ideally but not necessarily as it will be briefly presented: some familiarity with the material of Ch. 9, and 10.1-2 of Cycon-Froese-Kirsch-Simon.

Eugenia Malinnikova, Norwegian University of Science & Technology

Frequency function of solutions to second order elliptic equations and zero sets of Laplace-Beltrami eigenfunctions

In 1966,  Shmuel Agmon introduced the method of logarithmic convexity for weighted norms of solutions of second order equations. These ideas were developed by Almgren and later by Garofalo and Lin, in particular,  to prove unique continuation results for a wide class of second order elliptic equations. We will  give an introduction to Almgren’s frequency function, monotonicity formula and quantitative unique continuation results, including the three-ball inequality and Cauchy uniqueness inequality. These ideas can be applied to estimates of the zero sets of eigenfunctions of the Laplace-Beltrami operator on compact manifolds. In dimension two,  the first result was obtained by Donnelly and Fefferman in 1980s.  Their estimate for the size of the zero set from above was slightly improved in our recent work with A. Logunov, the new ingredients of the proof being the combinatorial properties of the frequency function. More detailed analysis of the frequency function led Logunov to new estimates for the zero set of eigenfunctions in higher dimensions, we will survey some of his results.

Prerequisites:  Real analysis and basic theory of elliptic PDE, some familiarity with Hausdorff measure.

Aaron Naber, Northwestern University

Effective Reifenberg theory with applications to singularity analysis for nonlinear equations

We shall discuss in this course two topic threads. First, we will go over recent results in effective reifenberg analysis.  Roughly, given a general measure whose support is approximately contained in a subspace for all points and scales (which is measured precisely by the Jones beta-numbers), we will see that such measures are automatically upper Alhfors regular and rectifiable. If time allows we will also study similar results for measures in Hilbert and Banach spaces (the latter is particularly subtle). The second part of the course will focus on applications, where we will see these assumptions appear in very real scenarios when studying the singular sets of nonlinear equations. We will focus our attention on nonlinear harmonic maps, which requires the least background, but this also occurs in minimal surfaces, yang-mills theory, free boundary problems, Ricci curvature, etc... In combination with new classes of estimates for the nonlinear equations, so-called L2 subspace approximation theorems, one is able to apply the effective reifenberg results to conclude rectifiable and measure bounds on the singular sets of such equations.  We will cover many of the basics of nonlinear harmonic maps, including stationary equations, monotone quantities, and the classical stratification, and attempt to work our way up to the newer results.

Prerequisites: For the first part I will attempt to keep the prereq's to basic measure theory, however some familiarity with Hausdorff measure, minkowski content, and the classical Reifenberg result would be helpful in facilitating an intuition. For the second part one should try and have some familiarity with the standard linear laplace operator on Euclidean space, and at least a cursory knowledge of linear elliptic theory. Though you will not have to have it, a set of Park City course notes by Leon Simons on nonlinear harmonic maps gives an excellent background to the nonlinear theory, which would be helpful knowledge in order to get the most from the course.

Zhongwei Shen, University of Kentucky

Boundary Layers in Periodic Homogenization

In this short course we will present some of the recent development on homogenization of second-order elliptic equations with rapidly oscillating periodic coefficients. We will start with an introduction of the classical homogenization theory and then study the quantitative properties of solutions: convergence rates and uniform regularity estimates. Finally, we describe recent work on the analysis of boundary layers for solutions of boundary value problems with oscillating boundary data.

Lecture I   Introduction to homogenization of elliptic equations

Lecture II   Convergence rates

Lecture III   Uniform regularity estimates

Lectures IV & V   Boundary value problems with oscillating boundary data

Prerequisites: A year-long graduate course on PDEs and a year-long graduate course on real analysis.

Charles Smart, University of Chicago

Stochastic Homogenization

I will discuss the homogenization of first and second order equations with random spatial dependence.  This will include first passage percolation, random surfaces, and random conductances.

Prerequisites. Students should be familiar with the basic elements of probability theory. For example, they should know about sigma-algebras, Martingales, and Azuma's inequality. Some familiarity with partial differential equations, especially the basic theory of divergence form equations, will be helpful in the second part of the course.