
LMS/EPSRC Short Course on Euclidean Harmonic
Analysis
April 1015, 2005
Organiser: Dr. James Wright


Introduction
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 fivehour
courses on
Introduction To
Euclidean Harmonic Analysis,
Nonlinear Hyperbolic Partial Differential
Equations and
Model Problems Over Finite Fields,
together with a threehour 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.
Programme
Courses IIII 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 minibus 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 onehour 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 IIII 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\'onZygmund theory of
singular integrals, LittlewoodPaley theory, Fourier multipliers,
maximal functions, and issues of pointwise convergence.
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.
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 BochnerRiesz, Restriction and Kakeya.
These problems address fundamental phenomena; BochnerRiesz 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 BochnerRiesz 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
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.
Registration
The registration fee is 100 pounds which, for UKbased research
students, includes the cost of accommodation.
Participants must pay their own travel costs.
EPSRCsupported 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
J.R.Wright@ed.ac.uk.
Page last modified: 05 April, 2005