Algebraic geometry and commutative algebra

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One of the amazing features of pure mathematics research is how shockingly often it discovers beautiful general results which can be used to help answer important problems in other parts of science and engineering. This is the case with the problems being considered in this proposal. The research is in the area called "algebraic geometry", a subject that has been around since the ancient Greeks studied the sections of a cone. The problem we consider in this proposal is: how big is the set of solutions to a collection of equations? Knowing how to answer that question for the particular equations that come up in this proposal will help decide: if an evolutionary tree proposed by biologists is correct; if it is possible to use an enormously powerful computer to multiply exceedingly large matrices in an efficient manner (something, for example, that economists are interested in doing when they seek to model our economy); how medical researchers can extract useful data from millions of patient records where each individual record contains either none or only a very little bit of data on one specific disease. One of the goals of the proposal is to train young researchers to become expert in algebraic geometry and to also understand the nature of the problems in biology, statistics and computing which are amenable to solution by the methods of this proposal. This is a new field of research in Canada which has the potential to offer opportunities to forge new connections between pure mathematics, biology, computing and statistics. My students will also have the opportunity to participate in the proposed European-based Research Network on Applications of Algebraic Geometry and other initiatives being planned (e.g. at the Mittag-Leffler Institute in 2010) which are connected to the research in this proposal.