Colorings of a graph, order preserving maps into a chain, and points outside a hyperplane arrangement over a finite field are just some examples of combinatorial structures whose counting functions are polynomials. A combinatorial reciprocity theorem is a form of duality that relates combinatorial structures via their counting functions. Informally: The counting function of the one is the counting function of the other at negative integers. In this course we will view such reciprocities from the perspective of geometric combinatorics. To that end we will develop the necessary theory of partially ordered sets and Moebius functions, polytopes, hyperplane arrangements, triangulations, and Ehrhart theory.
This course is aimed at beginning graduate/master students with an interest in combinatorics and discrete geometry. The course is part of the module "Discrete Geometry"
A solid background in linear algebra and some geometric intuition is sufficient but more mathematical background is surely welcome.
For more details see: Combinatorial Reciprocity Theorems Sommer 2015