Recent progress in Mathematics and Statistics: Khovanskii -- Algebraic Geometry and Convex Geometry

Askhold Khovanskii: Algebraic Geometry and Convex Geometry

Abstract: 

Newton polyhedra relate algebraic geometry and singularity theory with the geometry of 
convex polyhedra within the framework of toric geometry. This connection is useful in 
both directions. On the one hand, it provides explicit answers to problems in algebra 
and singularity theory in terms of convex polyhedra. For instance, according to the 
Bernstein-Khovanskii-Koushnirenko (BKK) theorem, the number of solutions of a generic 
system of n equations in (C^*)^n with  fixed Newton polyhedra is equal to the mixed 
volume of the Newton polyhedra multiplied by n!. This suggests that there should be an 
analog of the famous Alexandrov- Fenchel inequalities from the theory of mixed volumes 
in algebraic geometry. (These inequalities can be considered as a broad generalization 
of the classical isoperimetric inequality.) On the other hand, algebraic theorems of a 
general nature (such as the Hirzebruch-Riemann-Roch theo- rem) suggest unexpected results
 in the geometry of convex polyhedra. The theory of Newton-Okounkov bodies connects 
algebra and geometry in the broad framework of general algebraic varieties. This 
relationship is useful in many directions. It suggests the existence of birationally 
invariant theory of intersection of divisors and provides elementary proofs of 
Alexandrov-Fenchel inequalities in the theory of intersections and their local versions 
for the multiplicities of intersections of ideals in local rings. Alexandrov-Fenchel 
geometric inequalities easily follow from their algebraic analogs. In the theory of 
invariants, this connection gives analogues of the BKK theorem for horospherical 
varieties and some other varieties with the action of a reductive group. In abstract 
algebra, this relationship allows us to introduce a broad class of graded algebras, the 
Hilbert functions of which are not necessarily polynomials for large argument values, 
but have polynomial asymptotics. In my presentation, I will introduce these results in 
a way that is accessible to a general mathematical audience.

We plan to take the speaker to dinner so if you intend to join us please let me know by 
the next Monday.

Refreshments will be served in the Tea Room before the talk starting with 3:15 pm.