Tensor Decomposition - an algebraic and geometric approach

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The proposal for the research described concerns problems in the general area of tensor decomposition i.e. for a given tensor one is asked to find the minimal way to represent that tensor as a sum, using members of some family of simpler tensors. This elementary sounding statement actually contains many interesting and perplexing problems, not only in mathematics, but in physics, biology, statistics, computational complexity theory and communication theory. The wide range of applications for solutions to these problems has not only stimulated mathematicians to find solutions, but the applications themselves have suggested new and interesting mathematical approaches to these problems. This cross fertilization makes the area a rich and interesting place to put one's effort and encourages cooperation between scientists who are not always used to having important things to say to each other. To have e.g., statisticians, algebraic geometers and engineers working on the same problems is, to me, very exciting. It is anticipated that the research results in this area will increase our understanding of these fundamental problems and provide profound insight into their very nature. Coming at these problems from a variety of directions cannot but help us find new ways to think about what the solutions should be like. The variety of the approaches and the vastness of the problems makes this also an excellent area for researchers of every level to study. As I mention in the proposal, there is room for undergraduates, doing experiments with the problems, to Postdoctoral fellows, searching for patterns in the nature of the solutions. As for the impact that this work might have in Canada, I believe, that like most of mathematical and scientific research, the impact is worldwide since the audience does not have national boundaries. That being said, if solutions to these problems are found in Canada this will reflect very positively on our stature as a place to do research and to work with people who are finding answers to important questions. When a Canadian wins a Nobel Prize, the results of that research do not only help Canadians but the results do say that Canada is a place where excellent work is being done.