Monday, July 3, 2017
Technische Universität Berlin
Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin
room MA 041
Lecture - 14:15
Abstract:
Given a real closed polytope P, we first describe the Fourier transform of its indicator function 1 P by using iterations of Stokes' theorem. This idea has been tried before in various ways, and here we have a new way of handling the non-generic frequency vectors that arise in the Poisson summation formula, after taking the Fourier-Laplace transform of the indicator function 1 P .
We give an algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of P. The combinatorics of the face poset of P also plays a central role in the description of the Fourier transform of P. Also, we obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. This closed form generalizes known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.
Colloquium - 16:00
Abstract:
The notion of Ehrhart tensor polynomials, a natural generalization of the
Ehrhart polynomial of a lattice polytope, was recently introduced by
Ludwig and Silverstein. We will take a look at its coefficient and
introduce h*-tensor polynomials, extending the notion of the Ehrhart
h*-polynomial. Parallels as well as differences to the classical Ehrhart
Theory will be discussed. In particular, it is well known by Stanley's
Nonnegativity Theorem that the h*-polynomial has nonnegative coefficients
and generalizing this concept to h*-tensor polynomials will lead to
interesting results and open problems. Moreover, a tensor variant of
Hibi's palindromic theorem will be presented.
This is joint work with Katharina Jochemko (KTH Stockholm) and Laura
Silverstein (TU Vienna).