Focused Research Group

(National Science Foundation)

Goals and Principles • Principal Investigators • Lecture Series • Graduate Students & Postdoc Fellows • Upcoming Events

 




Article about our new FRG grant!

Click here for photos from the FRG seminar!


Goals and principles

The main goal of this group is to bring the diverse techniques of harmonic analysis, partial differential equations, additive number theory and geometric combinatorics together for the purpose of creating deeper and more interesting mathematics. The graduate student and postdoctoral participants will be exposed to the environment where artificial distinctions between different areas of mathematics will not seem reasonable or plausible.

We will keep a Report Page where shall describe the impact of the FRG activities that have already taken place.

Principal investigators and co-principal investigators

Steve Hofmann (University of Missouri)
Alex Iosevich (University of Missouri)
Michael Lacey (Georgia Institute of Technology)
Akos Magyar (University of Georgia)
Dorina Mitrea (University of Missouri)
Marius Mitrea (University of Missouri)
Gerd Mockenhaupt (Georgia Institute of Technology)

FRG lecture series

Mihalis Kolountzakis, Sergey Konyagin, Izabella Laba, and Michael Rudnev will deliver a series of lectures on topics related to the themes of the proposal.

Graduate Student and Postdoctoral Researchers

Georgiy Arutyunyants (MU), Simona Barb (MU), Mikhail Ganichev (MU), Derrick Hart (GT), Tunde Jakab (MU), William McClain (GT), Seick Kim (MU and MSRI), Doowon Koh (MU), Svetlana Mayboroda (Australian National University and Ohio State University), Steven Senger (MU), Qiang Shi (MU), Erin Thibault (MU), Dwight Thieme (MU), Nicholas Wegman (MU), Michael Williams (MU), and Mathew Wright (MU) are expected to participate in the FRG activities. This list is by no means exhaustive and we expect others to join as well.

2005-2006 Program of Events

Late August 2005

A series of lectures on analytic and combinatorial aspects of distance set theory by Michael Rudnev of the University of Bristol. A detailed description of the lectures will become available on this page in the near future.

Fall 2006 Graduate Course

Topic: Harmonic Analysis Techniques in Partial Differential Equations

Instructor: Marius Mitrea

Description: We will be concerned with the study of general elliptic boundary value problems under minimal smoothness assumptions on the domain and the coefficients of the operator involved. The goal is to develop the necessary tools and techniques which allow us to treat such problem for data in arbitrary Sobolev and Besov spaces. These include:

Winter 2006

FRG seminar. This is the continuation of the FRG seminar we have been running.

Winter 2006 Graduate Courses

Title: Topics in Harmonic Analysis and PDE:  Applications of Tb Theorems in PDE

Instructor: Steve Hofmann


Description: The course will concentrate on so-called Tb theorems and their applications to PDE.  Such theorems (first conjectured by Yves Meyer, and first proved in a certain special case by Meyer and McIntosh) originally arose in connection with an attempt to better understand the Cauchy integral operator on Lipschitz curves. The circle of ideas related to these theorems and their proofs has
subsequently found application to various problems in harmonic analysis and PDE, including the solution of the square root problem of Kato, the solution of Vitushkin's conjecture on analytic capacity, and the development of the layer potential method for divergence form elliptic operators with bounded measurable coefficients.

Here is a rough outline of topics that we plan to cover:

1. T1 Theorem (David and Journe)
2. Simple Tb Theorem for square functions (S. Semmes)
3. Full Tb Theorem (David-Journe-Semmes)
4. Cauchy integral via Tb Theorem
5. Solution of the Kato square root problem
6. M. Christ's ``local" Tb Theorem and extensions (Auscher, Hofmann,
   Muscalu, Tao, Thiele).
7. Application of local Tb Theorems to layer potentials for divergence form
   elliptic equations with non-smooth coefficients.

and, time permitting, we *may* also discuss *some* elements of the following:

8. uniform rectifiability, Mattila-Melnikov-Verdera
   Theorem, local Tb Theorems and analytic capacity (M. Christ), non
   doubling Tb Theorems (Nazarov, Treil, Volberg) and the
   solution of Vitushkin's conjecture (by G. David in the case of
   finite 1-D Hausdorff measure, and by Tolsa in general).

Prerequisites:  The rudiments of classical harmonic analysis of the Calderon school (basic theory of the Fourier transform, Hardy-Littlewood maximal function, approximate identities, Littlewood-Paley Theory, classical Calderon-Zygmund theory, BMO and Carleson measures).

Title: Arithmetic Progressions in the Primes and Fourier Analysis

Instructor: Alex Iosevich

We will go through Bryna Kra's description of Green and Tao seminar result on arithmetic progressions in primes. In the process we will have to learn some ergodic theory and analytic number theory.

Title: Harmonic Analysis and Partial Differential Equations

Instructor: Dorina Mitrea

The aim is to study issues such as regularity, boundary behavior, estimates, and integral representation formulas for solutions of various classes of PDE's which include the Laplace operator, the Lame system (of elastostatics) and the Stokes operator (of hydrodynamics). The emphasis is on the tools, typically from harmonic and functional analysis, which allow us to obtain optimal results.

Contents include:

  1. Hardy-Littlewood Maximal Function
  2. Singular Integrals, Maximal Operators, Cotlar's inequality
  3. Existence of pointwise values for singular integrals
  4. Fundamental solutions and layer potentials
  5. Jump relations
  6. Reduction of a boundary-value problem to a boundary integral equation
  7. Well-posedness in various smoothness spaces of basic problems from mathematical physics

March 2006

An FRG conference with emphasis on the interaction between harmonic analysis and partial differential equations. Detailed information about the conference will soon become available at http://www.math.missouri.edu/~iosevich/frgconference.html.

April 2006

A series of lectures by Sergei Konyagin. Topic: TBA

May 2006

A series of lectures by Izabella Laba. Topic: TBA