Analysis Seminar
Speaker: Kiumars Kaveh (University of Pittsburgh)
"Convex bodies, algebraic equations and group actions"
Time: 15:30
Room: MC 108
We discuss a (new) connection between algebraic geometry/representation
theory and convex geometry. We explain a basic construction which
associates convex bodies to semigroups of integral points. We see how this
gives rise to convex bodies associated to algebraic varieties encoding
information about their geometry. This far generalizes the notion of
Newton polytope of a toric variety. As an application, we give a formula
for the number of solutions of an algebraic system of equations
(equivalently self-intersection of a divisor/linear system) on any
variety, in terms of volumes of these bodies. This has many interesting
applications in algebraic geometry, in particular theory of linear
systems. We will see how several convex polytopes naturally appearing in
representation theory (of Lie groups) are special cases of this geometric
construction. The origin of this approach goes back to influential work of
A. Okounkov on multiplicities of representations.
Mathematics Calendar 
June 2011
