Date: Friday, November 18, 2011
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: Algebraic and convex geometries
Abstract: Traditionally, the connection between convex geometry and algebraic geometry has been restricted to the framework of toric varieties and Newton polyhedra. The theory of NewtonOkounkov bodies, developed in recent joint work with Kiumars Kaveh, transcends these limitations. Generalizing the notion of a Newton polyhedron, we define NewtonOkounkov bodies for semigroups of integral points, graded algebras and linear series on varieties. We show that for a large
class of graded algebras, the Hilbert functions have polynomial growth, and their growth coefficients satisfy a BrunnMinkowski type inequality. If time permitted we will also discuss a new version of the intersection theory. NewtonOkounkov bodies together with this intersection theory allow to prove an algebraic analogues of AlexandrovFenchel inequality and to give elementary proof of the classical AlexandrovFenchel inequality. These results provide an interplay between algebraic and convex geometries, and suggest a new geometric AlexandrovFenchel type inequality for mixed covolumes of convex bodies inscribed in a given convex cone.
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Speaker: Askold Khovanskii
Institution: U. Toronto
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