Conférence AndréAisenstadt Lecture
(Cette conférence s’adresse à un large auditoire. /
Suitable for a general audience.)
Le mardi 11 avril 2006 / Tuesday, April 11, 2006
16 h / 4:00 p.m.
Pavillon RogerGaudry, Université de Montréal,
Salle / Room M415
The representation of integers by quadratic forms
The classical Four squares theorem of Lagrange asserts that any positive integer can be expressed as the sum of four squares; that is, the quadratic form a^{2}+b^{2}+c^{2}+d^{2} represents all (positive) integers. When does a general quadratic form represent all integers? When does it represent all odd integers? When does it represent all primes? We will show how all these questions turn out to have very simple and surprising answers. In particular, we will describe the recent resolution, in collaboration with Jonathan Hanke (Duke U.), of Conway's 290conjecture.
Une réception suivra la conférence au Salon Mauricel'Abbé, Pavillon AndréAisenstadt (Salle 6245). / There will be a reception after the lecture in Salon Mauricel'Abbé,
Pavillon AndréAisenstadt (Room 6245).

Conférences dans le cadre du Séminaire
QuébecVermont Number Theory /
Lectures at the QuébecVermont Number Theory Seminar
Jeudi 27 avril 2006 / Thursday, April 27, 2006
Université de Montréal, Pavillon AndréAisenstadt, Salle / Room 5340
10h30 12h00
An overview of Gauss composition and its generalizations
We present 14 higher analogues of Gauss composition.
14h15  15h45
The parametrization of rings of low rank
A ring of rank n is a commutative ring with identity that is free of rank n as a Z. The prototypical example of a ring of rank n is, of course, an order in a degree n number field. How can one explicitly describe all rings of rank n for small values of n? The answer plays an important role in developing the composition laws of Lecture I, and also in understanding the distribution of number fields that will be dicussed in Lecture III.
Mardi 2 mai 2006 / Tuesday, May 2, 2006
Université de Montréal, Pavillon AndréAisenstadt, Salle / Room 5340
10h30  12h00
Counting field extensions of the rational numbers
A folk conjecture, possibly due to Linnik, states that the number of number fields of degree n and absolute discriminant less than X is ~c_{n}X for some constant c_{n}, where c_{n}> 0 for n >1. We show how the parametrizations described in the previous lectures, along with geometryofnumbers arguments, can be used to resolve Linnik's conjecture for n≤ 5.
Jeudi 4 mai 2006 / Thursday, May 4, 2006
Université de Montréal, Pavillon AndréAisenstadt, Salle / Room 5340
10h30  12h00
Mass formulae for local fields, and global heuristics
We present heuristics for the expected number of S_{n}number fields of given degree n and discriminant D. These heuristics allow us to derive explicit values for the constants c_{n} for all n, which agree with the values of c_{n} already established when n≤ 5. At the heart of these heuristics is a mass formula, inspired by work of Serre, that counts all deg n étale extensions of a local field.
