An undergraduate seminar during the fall of 2018.
A basic problem in combinatorics—in fact, probably in life!—is to count: how many objects are there of a particular shape, perhaps limited by a parameter \(n\)? Varying the parameter \(n\) gives a sequence of non-negative integers which can be packaged into a counting function. Remarkably often, these counting functions turn out to be restrictions of polynomials, or other nice functions defined on the entire set of integers, to the non-negative ones. One might ask: do evaluations of the counting function at negative integers mean something? Again, remarkably often, these negative evaluations turn out to count some sort of “dual” object. Such an answer is called a combinatorial reciprocity theorem.
With combinatorial reciprocity theorems as our guiding stars, this seminar is an introduction to the beautiful landscape of enumerative and geometric combinatorics. The gadgets and tools we will find at hand are those of modern algebraic combinatorics, such as the theory of partially ordered sets, Möbius functions, and generating functions. The local wildlife consist primarily of of polyhedra, polytopes, cones, and hyperplane arrangements, all of which have been marvelled since the ancients.
This seminar should be accessible to one with a good background in linear algebra.
Beyond the mathematical content, these seminars are really about communicating mathematics. This primarily means two things for me: First, the small seminar environment is a perfect place to develop skills and a sense of comfort in speaking and presenting mathematics. Thus the seminar consists mostly of lectures by the attendees. Second, I would like to try to give direct feedback on mathematical writing, so expect an occasional problem set which, in addition to having fun with the mathematical objects at hand, is a time to think about how to write mathematics.
Our primary reference is a beautiful new book by Mathias Beck and Raman Sanyal:
The idea of a combinatorial reciprocity theorem was probably first formalized by Richard Stanley in a 1974 paper where he thought deeply about certain algebraic identities amongst counting functions. His original paper is not an easy read, but it’s good to know where your roots lie.
Along the way, we are going to pick up some tools from algebraic and enumerative combinatorics. A classic reference in this area is Stanley’s books on enumerative combinatorics, only the first of which should be necessary.
A large part of our adventure will be centred around polytopes, triangulations, and other topics in discrete geometry. I think the delightful book by Mathias Beck and Sinai Robins on lattice point enumeration and Ehrhart theory should suffice for most of our purposes.
For more on discrete geometry, here are three classics:
We meet Monday and Wednesdays in Mathematics Room 528 between 4:30PM and 5:30PM.