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In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Gröbner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plücker relations. It is shown to parametrize all tropical(More)
The topic of this lecture was the tropical approach in mathematics. This approach was in its infancy at that time, but it has since matured and is now an integral part of geometric com-binatorics and algebraic geometry. It has also expanded into mathematical physics, number theory, symplectic geometry, computational biology, and beyond. We offer an(More)
This dissertation is dedicated to the memory of my mother, Kathleen Marie Harter. ii ACKNOWLEDGEMENTS I have come this far largely because I have had a great deal of support, encouragement , and companionship from amazing teachers, friends, and colleagues, and I would like to take this opportunity to thank some of those people. Graduate school at Michigan(More)
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