LMS/EPSRC Short Instructional Course
DIFFERENTIAL GEOMETRY, HOMOGENEOUS SPACES AND INTEGRABLE
SYSTEMS
University of Durham, 16-20 September 2002
GENERAL INFORMATION
The main purpose of this course is to provide training for
postgraduate students wishing to know about differential
geometry and related areas. As such, it will be of interest
to graduate students in pure mathematics, and
students in theoretical physics would also benefit.
The Course is open to all graduate students, and no
specialised postgraduate knowledge will be assumed.
Some undergraduate knowledge of some topics from
topology, linear algebra and calculus of
several variables will be required,
but there will be an opportunity for their review
on the first day of the Short Course.
The main theme of the Course will be differential geometry, and
our overriding objective is that the Course should feed the
participants' appetites for geometrical mathematics. We
have three main aims in mind. Firstly,
a student with only a very limited previous exposure to
differential geometry will attain a reasonable competence
in this area and also get some idea of the
many applications of the subject. Secondly, a student with
a good initial knowledge will be given a significant
insight into some areas of current research
activity. Thirdly, students should interact with
themselves and the speakers and tutors.
The Course consists of three intensive series of lectures,
each at the rate of one hour per day, from Monday 16th to
Friday 20th September 2002. The material will be accessible
to first year PhD students.
Problem sheets will be provided by the lecturers, and
tutorial classes will be given by post-doctoral tutors to help
students understand the material in depth.
One of the three intensive courses will be specifically
on differential geometry. There will be another one on
Lie groups and homogeneous spaces, and the third will
consider surfaces and integrable systems. Geometrical
aspects will be emphasised throughout.
In differential geometry, properties of geometric
configurations are investigated by means of differential
and integral calculus.
Applications of differential geometry can be found in
many branches of mathematics and modern sciences,
in particular in theoretical physics.
Lie groups,
homogeneous spaces and symmetric spaces are fundamental
mathematical objects and their general theory is already
well established. One of the basic features here is that many
geometric and analytic questions
can be tackled with algebraic methods.
A thorough
understanding of this theory is vital in many areas of
mathematics. Applications may
be found, for instance, in the modern theory of integrable
systems, in twistor constructions and
in submanifold geometries.
Surface theory has been a traditional topic in
differential geometry for many years. The modern
theory of integrable systems
has had a great impact on the development of this
topic in recent years. Old problems, thought to be
unrelated to each other, suddenly appear as the same
problem but just from different viewpoints. The
interplay between the theory of integrable systems
and classical surface theory is one of the most
fascinating areas in differential geometry at present.
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Last update: 7 March 2002