**Contents**: The school
intends to provide (prospective) beginning Master students in pure
mathematics with self-contained introductory courses in a range of
different topics. Our goal is to create a stimulating atmosphere
and
opportunities for an interchange of ideas, with each other and with the
lecturers. This year's edition offers three self-contained introductory
courses, each consisting of lectures and exercise sessions. These will
be supplemented by various talks by experts in related fields, to
demonstrate how the topics are related, and which research areas can be
found at our department. The three courses offered are:

- QRT
and Elliptic Surfaces (Hans Duistermaat):

A QRT map is a birational transformation from the plane to itself which has a very elementary definition in terms of a so-called pencil of biquadratic curves in the plane. Each of the biquadratic curves, when non-singular, is an elliptic curve, and the QRT map acts as a translation on it. Each point of the plane is contained in exactly one curve of the pencil, except for the so-called base points, which lie on every one of the curves. Isn't that remarkable? Blowing up the base points, one obtains a surface which is fibered by the curves, and because the non-singular ones are elliptic curves, the surface is called an elliptic surface. The QRT map defines an automorphism of the elliptic surface, which explains the title of the course. Application of methods from complex algebraic geometry to the automorphism of the elliptic surface leads to a very detailed understanding of the QRT map. Details can be found in my forthcoming book on this subject. In the course I will explain the QRT map, elliptic curves, and why in the complex projective setting every non-singular biquadratic curve is an elliptic curve. I will also explain why the QRT map acts as a translation on every non-singular biquadratic curve. Furthermore, what blowing up is, and how this process makes the members of the pencil of curves disjoint, leading to the aforementioned fibering of the surface. Exercises and pictures will help in understanding what is going on. Although the subject of QRT maps is a very specialistic one in algebraic geometry, the methods which I explain are very basic and useful. In algebra, in geometry, and even in analysis. - Distributions
(Johan Kolk): An abstract can be found here.

- Lie
Algebras and Integrable Systems (Johan van de Leur):
In these lectures we will study the action of n x n-matrices on an
n-dimensional vector space V. There are two kind of actions, viz.
action of
all invertible matrices, i.e., the Lie group GL(n,C) action, and that
of
all n x n-matrices the Lie algebra gl(n,C) action. We introduce the
notion
of representation, weights, highest weight vector and GL(n,C) group
orbit.
We extend this action to the tensor product of V with itself. In the
study
of this we will show that this tensor product under the action of the
Lie
algebra decomposes into two invariant vector spaces, given by the
symmetric
and anti-symmetric tensors. By generalizing the anti-symmetric tensors
we
obtain the exterior algebra and the action of both the Lie algebra and
the
Lie group on this. An other way to describe this exterior algebra is to
introduce creation and annihilation operators. These operators
anti-commute
and form a Clifford algebra. We show that the GL(n,C) group orbit is
characterized by one single equation.
The next step is to let n go to infinity in a certain way. We obtain
some
obstructions that can be solved. There is again a natural action of
some
infinite dimensional Lie algebra and Group and there are creation and
annihilation operators that form a Clifford algebra. Again the group
orbit
(of some highest weight vector) can be described in terms of one single
equation. However there is a new feature that is not present in the
finite
dimensional case. There is a correspondence, called boson-fermion
correspondence, between certain elements in the Lie algebra, that form
a
Heisenberg algebra, and the creation and annihilation operators in the
Clifford algebra. Using this correspondence it is possible to describe
the
exterior algebra in a different way. The equations that describe the
group
orbit transform under this correspondence into a collection of
differential
equations and the action of the Lie group on the highest weight vector
gives
solutions of these differential equations. In this way one obtains the
so
called Kadomtsev-Petviashvili hierarchy of differential equations.
We assume some elementary knowledge of group theory and expect that you
know
what the tensor product is of two vector spaces. You do not have to
know
what a Lie algebra or Lie group is. The lectures are more or less
self-contained.

**Lecturers**: Prof. dr. J.J. Duistermaat, dr. J.A.C. Kolk, dr.
J.W. van de Leur

Arrival/Registration/Welcome:
A bed in a double room will be booked for you from Saturday 16 August
until Friday 29 August. You can pick up your key at the University
Museum (address: Lange Nieuwstraat 106, T: 030-2538008) on Saturday or
Sunday between 12.00 and 16.30. If you arrive after 16.30 then
unfortunately you will need to make other arrangements for the night
following your arrival!

Registration will take place on Monday 18 August in the morning
(details follow), in the lecture room (that is, room 611 in the
Mathematics building).

Schedule: TBA

Location: All lectures will
take place in room 611 of the Mathematics building.

Social Activities:

- Mid School Dinner: The USMS hosts a dinner during the weekend of 23 and 24 August.
- The
Erasmus Student Network organises a whole variety of activities in
which you may participate. Check the website
for details.

- It is possible to buy a card allowing access to the sports facilities on the campus. The card costs Eur 12.50 and gives you unrestricted use of the tennis, squash, fitness and beach volleyball facilities at Olympos from 1 July until 1 September.

**Computer access: **The Mathematics department has computer
rooms which close around 17.00. In addition there are computers at the
(adjacent) Minnaert Building and in the Buys Ballot Laboratory. These
rooms remain open on weekdays until 20.00 (room BBL-011B and
Minnaert-Sterrenzaal). Please note that during the weekend all
university buildings are closed.

**Target group**: Advanced undergraduates or beginning graduates
in mathematics, with a solid background.

**Fee**: Eur 800, including housing and all course materials.

**
Scholarships**: A
number of scholarships is available. You should state your request and
motivate your financial situation
on a separate sheet.

Dutch students: please note the simplified application procedure for Dutch students.

**Information and registration: **No longer possible, except for
people not in need of accommodation. Please email
summerschool@math.uu.nl if you still want to register.

- Homepage of all Summerschools of the
University of Utrecht with links to the homepages of each school;

- All Summerschools of the Science Department
- Homepage of
the Mathematics Summerschool (with link to the application form)