LMS/EPSRC Short Course on Euclidean Harmonic Analysis

April 10-15, 2005

School of Mathematics

University of Edinburgh

Organiser: Dr. James Wright

[Introduction] [Programme] [Course Overview] [Accommodation] [Registration] [Further Information]


Euclidean Harmonic Analysis has its roots in the theory of Fourier series which in turn has its beginnings in the mathematical modelling of heat flow and wave propogation. Since the 1950's a programme was initiated to free the theory of Fourier series from its one dimensional setting and develop results in higher dimensions with important applications to elliptic (and then parabolic) partial differential equations. The tools and techniques for this theory were developed throughout the 1960's and early 1970's and became powerful enough to address some fundamental questions about the Fourier transform which had been beyond the scope of previous methods. Throughout the 1970's these basic issues evolved into a coherent programme of core problems which have been pursued by a great number of mathematicians until the present day. Many new ideas and methods naturally arose during the pursuit of this programme and have now seen many interesting applications in combinatorics, number theory and nonlinear partial differential equations.

This short course is aimed at postgraduate students in mathematics and will provide an introduction to euclidean harmonic analysis, the links between central problems and applications to nonlinear hyperbolic partial differential equations. The course will consist of three five-hour courses on Introduction To Euclidean Harmonic Analysis, Nonlinear Hyperbolic Partial Differential Equations and Model Problems Over Finite Fields, together with a three-hour course on Central Problems In Euclidean Harmonic Analysis. The four invited lecturers are leading researchers in their respective fields and are widely known as enthusiastic and stimulating teachers.


Courses I-III will start on Monday morning at 9am in Lecture Theatre C which is located on the ground floor of the James Clerk Maxwell Building (JCMB) of the King's Building of the University of Edinburgh. For the participants residing in Pollock Halls there will be a mini-bus to take you to JCMB each morning, leaving at 8.30am. Course IV will start on Tuesday. Here is the timetable for the lectures
Timetable. Each course will consist of one-hour sessions and will comprise of formal lectures and tutorials in the form of examples classes. Five of the sessions will be given as lectures for courses I-III whereas three of the sessions will be given as lectures for course IV. The lecturers will distribute their tutorials flexibly within their programme. The lecturers will run the tutorials themselves to ensure that they get direct feedback from the students. Printed course notes and example sheets will be provided for all courses. Lunch will be served from 12.30 each day in Room 5215 JCMB except on Thursday (5327 JCMB). A banquet dinner will be held at Pizza Express on Thursday evening at 7pm (111 Holyrood Road , Edinburgh. Tel: 0131 557 5734).

Course Overview

Introduction To Euclidean Harmonic Analysis (Dr. Jonathan Bennett, University of Birmingham)

These lectures will develop fundamental material which underlies all of Euclidean Harmonic Analysis and will provide the context of the other lecture courses. There will be several examples discussed throughout and motivating remarks linking the material with other topics, such as those covered by the other lecturers.

Topics will include: the Fourier transform, Plancherel's Theorem, Fourier inversion, the action of the Fourier transform on functions in the Schwartz class, distributions, the method of stationary phase and the role of curvature, interpolation, the Calder\'on--Zygmund theory of singular integrals, Littlewood--Paley theory, Fourier multipliers, maximal functions, and issues of pointwise convergence.

Nonlinear Hyperbolic Partial Differential Equations (Dr. Nikolaos Bournaveas, University of Edinburgh)

Any hyperbolic equation is a wave equation and the solutions of such equations tend to be oscillations which spread out in space. A nonlinear term (such as u^p ) will tend to magnify the size of u when u is large and to be negligible when u is small. It can also make a solution blow up in finite time. In these lectures we will illustrate how modern techniques and phenomena from euclidean harmonic analysis (such as the restriction phenomenon; see Dr. Wisewell's lectures click) can be employed to prove existence and uniqueness of solutions to nonlinear hyperbolic partial differential equations as well as to study some qualitative properties of such solutions.

Topics will include: linear wave equation, energy estimates, Cauchy problem, existence and uniqueness. Sobolev spaces, embedding theorems. Semilinear wave equations, classical local existence theorem. The restriction theorem for the Fourier transform. Strichartz estimates. Local existence revisited (low regularity local solutions). Hardy's inequality. Global existence for wave equations with power nonlinearities via Strichartz estimates. Null form estimates. Wave maps.

Models Over Finite Fields (Professor Anthony Carbery, University of Edinburgh)

Many of the central problems of Euclidean Harmonic Analysis, as outlined in Dr. Wisewell's lectures click, seem to be a very long way off from seeing a complete solution. It has been clear that there are basic combinatorial and geometric issues underpinning these problems which we do not understand. Recently several people have attempted to see whether one can model some of the central problems in Euclidean Harmonic Analysis (where the underlying field is the real numbers) on vector spaces over a finite field. Surprisingly one does indeed have very interesting model problems which successfully highlight combinatorial issues we need to understand. In these lectures we will describe these models and explain how to translate Euclidean thinking to the setting of finite fields. Knowledge of finite field theory is not necessary.

Central Problems In Euclidean Harmonic Analysis (Dr. Laura Wisewell, University College, London)

Some of the central problems of Euclidean Harmonic Analysis are often referred to as Bochner-Riesz, Restriction and Kakeya. These problems address fundamental phenomena; Bochner-Riesz is concerned with inverting the Fourier transform whereas Restriction attempts to quantify which types of singularities the Fourier transform is allowed to possess. Kakeya is a geometric problem describing how well straight lines can be packed into space. These problems turn out to be intimately connected, a complete solution to Bochner-Riesz implies the complete solution of Restriction which in turn solves the Kakeya problem. In these lectures, we will describe these problems and explain their interconnections.


Accommodation has been arranged in single rooms with ensuite at
Pollock Halls . In addition to accommodation, breakfast and dinner will be provided at Pollock Halls for course residents. Lunch will also be provided for course participants. Please arrive at the Reception Centre of Pollock Halls (click on "Location map" under the Pollock Halls link above) after 2pm on Sunday, 10 April. Registration/Reception will begin at 4pm in the Ewing House Small Common Room (see "Location map"). The course will finish at 3pm on Friday, 15 April so arrange your travel back home sometime after this time.


The registration fee is 100 pounds which, for UK-based research students, includes the cost of accommodation. Participants must pay their own travel costs. EPSRC-supported students can expect that their registration fees and travel costs will be met by their departments from the EPSRC Research Doctoral Training Account.

The number of participants will be limited and those interested are encouraged to make an early application. An online application form is available from the London Mathematical Society.

The closing date for applications is 18 Feburary 2005.

Further Information

Further information is available from:

Dr. James Wright
School of Mathematics
University of Edinburgh
Mayfield Road
Edinburgh EH9 3JZ
Tel. +44 (0)131 650 8570
Fax. +44 (0)131 650 6553

Page last modified: 05 April, 2005