Independent sets and sphere packing: from graphs to high-dimensional geometry
ABSTRACT
The sphere packing problem asks how densely non-overlapping balls can be arranged in space. Despite its simple formulation, the problem becomes significantly more difficult as the dimension increases.
A useful approach is to translate this geometric problem into a combinatorial one: we represent points as vertices of a graph and connect pairs that are too close, so that a packing corresponds to an independent set. This perspective allows us to apply tools from graph theory and statistical physics, such as the hard-core model, to study packing density.
In this talk, I will introduce this framework and describe recent progress on lower bounds for sphere packing in high-dimensional Euclidean space.
I will also discuss extensions of these ideas to non-Euclidean settings, such as hyperbolic space.
