greg mcshane

Conway: Sensual quadratic forms

This page is a repo for exchanges in the gpe de lecture E.

IMPORTANT: this group will no longer meet physically because of confinement

We are meeting on Discord.

Background

The text is english so it might be useful to read some background material in english before the class.

There is a more recent text which has a lot of overlap with Conway’s book and we will be comparing the two versions when appropriate.

The theme of both books is the interplay between geometry and number theory.

We will spend 4 weeks on each of the first 3 chapters of Conway:

References for background reading

Follow the blue arrows -> for a reference.

For Wednesday 22/1 just read through Hatcher’s book


Plan for the seminars

Can you see the shape of a quadratic form

  1. Generalities on quadratic forms (definite, indefinite, integer valued, Pythagorean triples, Diophantine equations etc.)
  2. Farey diagram, SL(2,Z), Poincaré plane/disc (see Hatcher)
  3. Topography I
  4. Topography II
  5. Generalizations: Markoff numbers, Appolonian packings etc.

Can You Hear the Shape of a Lattice?

  1. Generalities on lattices, theta functions
  2. Milnor example, Kneser etc.
  3. Codes etc.
  4. Drums eigenfunction transplantation Buser, Berrard, Tsunada etc.

… and Can You Feel Its Form?

  1. Voronoi cells
  2. Vonorms
  3. Minkowski etc

Exposes

date nom sujet
5/2 Victor triplets
12/2 William sl2 et ses actions
19/2 Thomas topographie
11/3 Corentin descente
25/3 Vincent Markoff numbers
1/4 Milan formes modulaires et reseau
8/4 Vincent et Victor progressions arithmetiques

Further reading