Abstract:

In 1930, Keller conjectured that any tiling of $n$-dimensional space by translates of the unit cube must contain a pair of cubes that share a complete ($n-1$)-dimensional face. This generalized a 1907 conjecture of Minkowski in which the centers of the cubes were assumed

to form a lattice.

Perron proved Keller's conjecture to be true for $1 \leq n \leq 6$ in 1940, but Lagarias and Shor found a counterexample to Keller's conjecture in dimension 10 in 1992. I found a counterexample in dimension 8 in 2002. In this talk, I'll explain how Brakensiek, Heule and I recently used geometry and a SAT solver to rule out the existence of a clique of size 128 in the Keller graph $G_{7,3}$, thus showing Keller's conjecture to be true in dimension 7.

427 Thackeray Hall