LMS/EPSRC Short Course on Euclidean Harmonic Analysis
April 10-15, 2005
School of Mathematics
University of Edinburgh
Organiser: Dr. James Wright
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LMS/EPSRC Short Course on Euclidean Harmonic AnalysisApril 10-15, 2005School of MathematicsUniversity of EdinburghOrganiser: Dr. James Wright |
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| [Introduction] | [Programme] | [Course Overview] | [Accommodation] | [Registration] | [Further Information] |
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.
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.
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.
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.
Dr. James Wright
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.
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
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 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.
Further Information
Further information is available from:
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.