Type: Seminar

Distribution: World

Expiry: 10 Aug 2023

CalTitle1: Algebraic Geometry and Convex Geometry

Auth: laurent@p721m2.pc (assumed)

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 â€€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.

Calendar (ICS file) download, for import into your favourite calendar application

UNCLUTTER for printing

AUTHENTICATE to mark the scnews item as read