Toward a Classification of the Ranks and Border Ranks of All (3,3,3) Trilinear Forms

The study of the ranks and border ranks of tensors is an active area of research. By the example of determining the complexity of matrix multiplication I introduce the reader to the notion of the rank and border rank of a tensor. Then, after presenting basic preliminary material from algebraic geometry and multilinear algebra,I quantify precisely what it means for some tensor to be of given rank, border rank,symmetric rank or symmetric rank. Objects of a given (symmetric) border rank are then interpreted geometrically as elements of certain secant varieties of Veronese and Segre varieties. Using this, I describe some of the techniques used to arrive at the classification of all (3,3,3) presented by Kok Omn Ng. The main result of this thesis is a classification of all the border ranks and some of the ranks of the 24 normal forms given by Kok Omn NG in "The Classification of (3,3,3) trilinear forms"