Georgia Institute of Technology College of Sciences

We present a lemma, inspired by dependent random choice and sampling procedures from statistical physics, for finding dense structure in arbitrary $d$-partite $d$-uniform hypergraphs. We will then discuss how this lemma leads to the concept of local rank, a notion of tensor rank which is instrumental in proving a "structure vs. randomness" result for tensors (and by extension, polynomials): namely, a relation between the partition and analytic ranks of tensors over finite fields. This is joint work with Guy Moshkovitz.