ABSTRACT: Given a convex body \(B\) which is embedded in
a Euclidean space \(\mathbb{R}^d\), we can ask how many lattice
points are contained inside \(B\), i.e. the number of
points in the intersection of \(B\) and the integer
lattice \(\mathbb{Z}^d\). Alternatively, we can count the lattice
points inside \(B\) 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.