|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 Newton-Okounkov bodies, developed in recent joint work with Kiumars Kaveh, transcends these limitations. Generalizing the notion of a Newton polyhedron, we define Newton-Okounkov 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 Brunn-Minkowski type inequality. If time permitted we will also discuss a new version of the intersection theory. Newton-Okounkov bodies together with this intersection theory allow to prove an algebraic analogues of Alexandrov-Fenchel inequality and to give elementary proof of the classical Alexandrov-Fenchel inequality. These results provide an interplay between algebraic and convex geometries, and suggest a new geometric Alexandrov-Fenchel type inequality for mixed covolumes of convex bodies inscribed in a given convex cone.
Speaker: Askold Khovanskii
Institution: U. Toronto