Orateur: Hester Pieters
Titre: The covering radius of the Leech lattice.
Résumé: The Leech lattice is remarkable for many reasons. Aside from its huge automorphism group, which is of great interest to group theorists, it turns up in number theory, coding theory, and theoretical physics. It was discovered in 1965 by J. Leech as the solution to the kissing number problem: how many spheres of equal size can touch a given one? The Leech lattice gives the unique solution in dimension 24. The only other dimensions where this question has been solved are 1,2,3,4 and 8. Soon after its discovery J. Leech conjectured that its covering radius is equal to $\sqrt{2}$. This was proved by J.H. Conway, R.A. Parker and N.J.A. Sloan in 1982. Their proof involved long calculations and case-by-case verifications. Later R.E. Borcherds gave a uniform proof by embedding the Leech lattice in a hyperbolic lattice of rank 26. I will discuss this proof which involves hyperbolic geometry and hyperbolic reflection groups.

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