  Motivated by problems in algebraic complexity theory (e.g., matrix
multiplication) and extremal combinatorics (e.g., the cap set problem and the
sunflower problem), we introduce the geometric rank as a new tool in the study
of tensors and hypergraphs. We prove that the geometric rank is an upper bound
on the subrank of tensors and the independence number of hypergraphs. We prove
that the geometric rank is smaller than the slice rank of Tao, and relate
geometric rank to the analytic rank of Gowers and Wolf in an asymptotic
fashion. As a first application, we use geometric rank to prove a tight upper
bound on the (border) subrank of the matrix multiplication tensors, matching
Strassen's well-known lower bound from 1987.
