31/05, IMPA. Sala: 232
Okounkov bodies and toric degenerations |
Resumo. Given a projective variety X of dimension d, a "flag" of subvarieties Y_i, and a big
divisor D, Okounkov showed how to construct a convex body in R^d that captures
interesting aspects of the geometry of D. In the last five years, this construction has
been developed further in work of Kaveh-Khovanskii and Lazarsfeld-Mustata. In general,
this "Okounkov body" is quite hard to understand, but when X is a toric variety, it is
just the polytope associated to D via the standard yoga of toric geometry. In this talk,
I'll describe recent results of myself and others relating Okounkov bodies on more
general varieties with flat degenerations to toric varieties, integrable systems, and
When is every nonnegative quadric a sum of squares?|
Resumo. If X is a subvariety of projective space we ask the following
question: When is every nonnegative quadric on X a sum of squares of linear
forms in X?
We identify a numerical invariant which is an obstruction for equality and
show that it vanishes iff X is a variety of minimal degree. These result
generalize a well known Theorem of Hilbert characterizing the pairs (d,k)
for which every nonnegative homogeneous polynomial of degree 2d in k
variables is a sum of squares and give us many new instance of the equality
of much interest in optimization.
These results are joint work with Gregoriy Blekherman (Georgia Tech) and
Gregory Smith (Queen's University).