250131 VO Topics in Combinatorics (2018S)

This course will be an introduction to integer-point enumeration in polyhedra: counting integer-points in polyhedra often leads to polynomial or quasi-polynomial enumeration formulas. Some people believe that the converse is also true: whenever we are given a counting problem whose counting function is a polynomial, the problem can be phrased as the problem of counting the integer-points in a certain family of polyhedra. We will develop this combinatorial theory and its connection to geometry and number theory. In particular, we will also study a phenomenon that is called combinatorial reciprocity: a priori, the counting polynomials that appear in connection with polyhedra only have a combinatorial interpretation for positive parameters, however, there are instances where we can give an interpretation also to negative parameters.