Beyond rank
Location
The notion of the rank of a matrix is one of the most fundamental in linear algebra. The analogues of this notion in multilinear algebra — e.g., what is the “rank” of an m x n x p array of numbers? — are much less well-understood, and are often thought of as of niche interest. At least, that’s how I was brought up to think of them, until Terry Tao explained to me that the resolution of the cap set conjecture by Croot, Lev, Pach, Gijswijt and myself really made use of these ideas! In fact, these notions are of great current interest in a wide range of mathematical subjects at the moment! Issues about “higher rank” arise in complexity theory, data science, geometric combinatorics, additive number theory, quantum mechanics, and commutative algebra — I willmanage to say something about some to-be-specified proper subset of these topics, and am happy to chat afterwards about the others.