## Lower bounds for kissing numbers

**Goal:**
The kissing number in dimension d is the maximal number of non-overlapping balls of radius one that can be tangent to a given ball of radius 1. It is obvious that the kissing number is 2 in dimension d=1, and not difficult to prove that it is 6 in dimension d=3. It is much more difficult to prove that the kissing number is 12 in dimension 3, and the exact values of the kiising number in dimensions 5 to 7 are not known. Kissing numbers are relevant in some applications (coding theory). Very elaborate methods have been designed and used to find upper bounds on kissing numbers in relatively small dimensions.

The first part of the project will be to use elementary geometric constructions to transform the definition of the kissing number in a way that involves configurations of points (centers of spheres) in Euclidean space of dimension d-1.

The second step will be to implement minimization algorithms (gradient descent, simulated annealing) to search for "best" configurations of points in dimension d-1.

The third step will be to try to find recover known lower bounds in small dimension, and possibly to improve on those which are known.

**Schedule:** to be determined.

**Persons involved:**
Advised by Jean-Marc Schlenker, NN.

**Difficulty level:**
Introductory.

**Tools:**
programs in python/sage, implementation of optimization algorithms to find good configurations of points on a sphere.

**Expected results:**
Construct configurations of points on spheres of dimension d-1 such that the minimum distance between two points is bounded from below and the number of points is a large as possible.