SUMMER SCHOOL IN GEOMETRY
23 - 29 JUNE 2002
CAMPUS SCIENTIFIQUE DE L'UNIVERSITÉ DE SAVOIE, LE BOURGET-DU-LAC, FRANCE
Hyperbolic manifolds and arithmetic
(an introduction via examples and constructions)
Organizers: G. Besson (Grenoble, France), O. Burlet (Lausanne, Switzerland), P. Buser (EPFL, Switzerland),
B. Colbois (Neuchâtel, Switzerland), F. Pelletier (Chambéry, France), P. Verovic (Chambéry, France).
People to be contacted: B. Colbois or P. Verovic.
The Summer School will be based on the three following courses each of which is six hours long:
Françoise Dal'bo (Rennes, France) - Introduction à une approche géométrique de l'arithmétique (will be held in French).
Alexander Mednykh (Novosibirsk, Russia) - Three-dimensional hyperbolic manifolds.
Alan Reid (Austin, USA) - Constructions of arithmetic hyperbolic manifolds.
The courses will be given in the morning and afternoons will be devoted to work and discussions in small groups for which the three speakers may be asked for more explanations.
The Summer School is particularly aimed at Ph.D. students and young researchers, but it is also open to more experienced mathematicians as far as possible.
There will be a maximum of 35 participants.
During the Summer School participants will be provided board and lodging in a student's home on the campus for a 65 Euro (CHF 100.-) registration fee. We can also book some rooms in a hotel for people who ask us for but the price difference will be on their own. On the other hand travel expenses are payable either by the participants or by their maths departments.
All the information about the Summer School location is available on the Web site of the Mathematics department (Lama) of the Université de Savoie (click on "Pour nous rendre visite").
Registration deadline: May 1st, 2002.
Everyone who would like to attend the Summer School is required to send an e-mail to P. Verovic with his/her name, nationality, professional information (occupation, postal address, e-mail and fax) and with his/her accommodation choice between a room in a student's home with full board (registration fee = 65 Euros or CHF 100.-) or a room in a hotel to be paid by the participant (less the price difference with the student's home accommodation).
- Courses description -
Our aim in this course will consist in pointing out the connections between diophantine approximations
and hyperbolic geometry.
In order to understand these links, we will focus on the modular surface before moving towards some generalizations.
J. Cassels - An introduction to diophantine approximation, Cambridge University Press.
S. J. Patterson - Diophantine approximation in Fuschian groups, Philosophical transactions of the Royal Society of London (series A, vol 282), 1976.
C. Series - The geometry of Markov numbers, Mathematical intelligencier (vol 7, n°3), 1985.
S. Herzonsky & F. Paulin - Diophantine approximation for negatively curved manifolds, preprint Orsay, 1999.
Part 1: Hyperbolic manifolds and orbifolds via polyhedra --- Coxeter polygons and polyhedra have been used to construct Riemann surfaces together with three-dimensional Euclidean, spherical and hyperbolic manifolds. The constructions are supposed to be done in two different ways: with pure geometric tools on the one hand and in terms of subgroups of discrete groups generated by reflections on the other hand. In particular, the Borromean rings orbifold can be obtain in a such way and non-convex polyhedra will be used to produce orbifolds for the figure eight, Hopf link and Whitehead link orbifolds.
Part 2: Branched coverings of low-dimensional manifolds and Reidemeister-Schreier method --- Regular and irregular branched coverings of Riemann surfaces are constructed by starting with simple polygons in the hyperbolic plane. The Reidemeister-Schreier method is used to find explicitly the subgroups of Fuchsian groups related with these coverings. The same idea will also apply in the three-dimensional case to describe manifolds as branched coverings over the three-sphere.
Part 3: Hyperbolic and spherical volumes of link and knot orbifolds --- Schlaefli formula will be introduced to calculate spherical and hyperbolic volumes of three-dimensional polyhedra and orbifolds. In a few simple cases such a calculation will be produced in a very explicit way with results that can be represented in terms of the integral form and the Lobachevsky function.
Arithmetic methods have been important in the development of constructions of discrete isometry groups of hyperbolic spaces. Indeed, although non-arithmetic manifolds exist in all dimensions, in dimensions at least 4 they consist essentially in just cutting and pasting arithmetic examples. We will here discuss constructions of arithmetic hyperbolic manifolds. In low dimensions this will involve developing some of the theory of quaternion algebras over number fields and their completions, and in higher dimensions we will use the theory of quadratic forms and their orthogonal groups. In dimension 3, we will also put these arithmetic constructions in the broader framework of how number theoretic methods arise in the study of hyperbolic 3-manifolds. Many examples and applications will be given throughout these lectures.
- Program -
Sunday 23 June is the arrival day.
The meeting point for all the participants will be in the maths building "Le Chablais" (see map on the Mathematics department Web site).
The courses will start on Sunday 23 June at 2:30 p.m.
N.B.: If for some travel reasons someone cannot get the Summer School on Sunday 23 June before 1:30 p.m., he/she will just have to tell us in advance he/she will arrive on Saturday 22 June and we will arrange about accommodation.
Click here for the day-by-day schedule.
- Accommodation -
Rooms in the student's home on the campus are furnished with a bed, a desk and a washbasin.
Toilets and bathrooms are in the corridor.
A kitchen for cooking is available on the same floor.
No phone in rooms but phone boxes next to the student's home. In case of emergency people can be called at the student's home reception.
- Facilities on the campus -
Internet access during the University working hours.
A post office and a travel agency.
Shops not available on the campus but in Le Bourget-du-Lac (nearest village located 500 m far from the campus).
- Activities -
Good walking schoes are needed for people interested in a walk we will organize in the area and
swim-suits are not to be forgotten for those who are planning to have a bath in Lake du Bourget.
Sport facilities are also available on the campus for playing basketball, volleyball, football, etc.
- Tourist information -
Tourist office in Chambéry.
Tourist office in Aix-les-Bains.
- Looking for more mathematics? -
International colloquium Vector bundles over algebraic curves from July 1st to July 5th, 2002 in Luminy (CIRM).
For further information about the Summer School please contact B. Colbois or P. Verovic.
Last update: March 5, 2002