Wednesday, March 30th, 2016
3:00pm - 4:00pm, in McCormack 2-404

Quang-Nhat Le

Brown University

Counting lattice points inside a dd-dimensional polytope via Fourier analysis

Abstract: Given a convex body BB which is embedded in a Euclidean space RdRd, we can ask how many lattice points are contained inside BB, i.e. the number of points in the intersection of BB and the integer lattice ZdZd. Alternatively, we can count the lattice points inside BB with weights, which sometimes creates more nicely behaved lattice-point enumerating functions. The theory of lattice-point enumeration in convex bodies is a classical subject that has been studied by Minkowski, Hardy, Littlewood and many others. For polytopes, the work of Ehrhart, Macdonald and McMullen in 1960s and 1970s has revealed many curious properties of various weighted and unweighted lattice-point counts of integer polytopes such as polynomiality and reciprocity laws. The enumerative theory of lattice points inside polytopes have found far-reaching applications in many mathematical areas such as toric varieties, symplectic geometry, number theory, mirror symmetry, etc. The use of Fourier analysis in the theory of lattice-point enumeration has recently enjoyed a renaissance that was pioneered by Barvinok, Brion & Vergne, Diaz & Robins, Randol and others. In this talk, we will investigate Macdonald's solid-angle sum of a polytope, which is a weighted lattice-point count with solid-angle weights. We will employ the Poisson summation formula and other combinatorial techniques to convert the calculation of the solid-angle sum to the computation of the Fourier transform of the polytope. Classically, the theory is concerned with integer dilates of integer and rational polytopes, but our methods are applicable to arbitrary real dilates of any real polytope. This is joint work with Ricardo Diaz and Sinai Robins.