Georgia Institute of Technology College of Sciences

How densely can one pack spheres in $d$-dimensional space? It is not too hard to show a lower bound of $2^{-d}$. (The only known upper bounds are exponentially larger.) Various proofs of lower bounds of the form $cd2^{-d}$ have been given; recently, Campos, Jenssen, Michelen, and Sahasrabudhe gave the first asymptotic improvement on such bounds in 75 years. I will discuss an extension of this improvement to packing other shapes in high dimensions, along with some connections to log-concave probability.