Title: Gaussian formulas for the number of integer points and volumes
of polytopes
Speaker: Alexander Barvinok
Abstract: We estimate the number of integer points and volumes of
higher-dimensional polytopes via solutions
to some particular convex optimization problems on the polytopes.
While we can prove the validity of the resulting
formulas in a number of interesting cases, preliminary computational
experiments suggest that the formulas
provide a very good heuristic for a much wider class of polytopes.
The optimization problems are essentially
the problems of maximizing the entropy of a distribution supported on
the set of non-negative integer points
or on the whole non-negative orthant with the expectation in a given
affine subspace and can be efficiently
solved by interior point methods. The talk is based on the joint work
with J. Hartigan, while computational data
are supplied by J. De Loera.