# Utrecht Summerschool in Mathematical Sciences (USMS) 2008

From August 18 to 29, 2008 the Department of Mathematics organises a Summer School entitled Topics in Algebra, Analysis and Geometry. Focus will be primarily on Analysis.

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:

Abstracts of each course:
• 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.