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.