Chaire Aisenstadt 2005-2006 Aisenstadt Chair
Manjul Bhargava
(Princeton University)

Conférence André-Aisenstadt Lecture

Photos de l'événement / Events' photos

(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 Roger-Gaudry, Université de Montréal,
Salle / Room M-415

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 a2+b2+c2+d2 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 290-conjecture.

Une réception suivra la conférence au Salon Maurice-l'Abbé, Pavillon André-Aisenstadt (Salle 6245). / There will be a reception after the lecture in Salon Maurice-l'Abbé,
Pavillon André-Aisenstadt (Room 6245).

Conférences dans le cadre du Séminaire
Québec-Vermont Number Theory /
Lectures at the Québec-Vermont Number Theory Seminar

Jeudi 27 avril 2006 / Thursday, April 27, 2006
Université de Montréal, Pavillon André-Aisenstadt, 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 André-Aisenstadt, 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 ~cnX for some constant cn, where cn> 0 for n >1. We show how the parametrizations described in the previous lectures, along with geometry-of-numbers arguments, can be used to resolve Linnik's conjecture for n≤ 5.

Jeudi 4 mai 2006 / Thursday, May 4, 2006
Université de Montréal, Pavillon André-Aisenstadt, Salle / Room 5340

10h30 - 12h00
Mass formulae for local fields, and global heuristics

We present heuristics for the expected number of Sn-number fields of given degree n and discriminant D. These heuristics allow us to derive explicit values for the constants cn for all n, which agree with the values of cn 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.

March 8 mai 2006