Title: Toward a Classification of the Ranks and Border Ranks of All (3,3,3) Trilinear Forms
Abstract: Toward a Classification of the Ranks and Border Ranks of All (3,3,3) Trilinear Forms. (April 2011) Derek James Allums Department of Mathematics Texas A&M University Research Advisor: Dr. J.M. Landsberg Department of Mathematics 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) trilinear forms 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.
Publication Year: 2011
Publication Date: 2011-08-08
Language: en
Type: dissertation
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Cited By Count: 1
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