Seminar Event Detail


Combinatorics

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.

Files:


Speaker:  Askold Khovanskii
Institution:  U. Toronto

Event Organizer:     

 

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