Colloquium Talks

I have developed several colloquium talks that I enjoy giving for undergraduate audiences. Below are descriptions of some of my favorites. I would love to give a talk for your students!

The Alternating Sign Matrix Conjectures

In a 1991 article in the Mathematical Intelligencer, David Robbins made the following proclamation.

These conjectures are of such compelling simplicity that it is hard to understand how any mathematician can bear the pain of living without understanding why they are true.
The conjectures Robbins was referring to concern the number of alternating sign matrices of various types, and so are sometimes called the alternating sign matrix conjectures. In this talk I explain what alternating sign matrices are, what the alternating sign matrix conjectures say about them, and the remarkable evidence Robbins and his colleagues found for believing them. One of the most striking of these conjectures was proved by Greg Kuperberg in 1995. Kuperberg was not the first to prove this conjecture, but his proof was certainly the simplest. I describe the surprising connection between alternating sign matrices and a model of ordinary ice which led to this simplicity. I also discuss a seemingly unrelated set of objects, called totally symmetric self-complementary plane partitions, which turn out to have a close and still mysterious connection with alternating sign matrices.

This talk is accessible to anyone who has seen matrices, determinants, and Pascal's triangle.
Stable Matchings: How Two Mathematicians used Hospitals and Residents, College Admissions Offices, and George Lopez's Kidney to Win a Nobel Prize in Economics

Matching problems are everywhere: high school seniors work hard to match themselves with colleges and universities around the country, medical students are matched with hospitals for their residencies, law students are matched with judicial clerkships, and a committee of doctors matches donated organs to transplant recipients. In these examples (and many others) we want to construct something called a stable matching. In particular, we do not want to have a college admissions process in which a student attending one college would prefer to be at the college across the river, while the college across the river would prefer that student to one of its own. In this talk I first describe a natural algorithm for constructing a stable matching from the preferences of the people or institutions being matched, and the advantages and disadvantages of variants of this algorithm. I then describe some remarkable structure on the set of all stable matchings (for a given set of preferences), which lets you "add" and "multiply" stable matchings.

This talk does not require any special background.

How Good is God at Counting Permutations?

In March of 2005 Doron Zeilberger declared,

Not even God knows a1000(1324).
In this talk I unpack Zeilberger's claim, describing what a permutation is, what it means for one permutation to avoid another, and explaining how studying this relationship leads to cool sequences like the Fibonacci numbers, the triangular numbers, and the Catalan numbers. This will lead us to some fascinating problems, some of which are open, and a bet between Zeilberger and Einar Steingrimsson which has not yet been settled.

No special background is required to enjoy this talk, but certain parts will be easier to understand if you have seen sequences and limits.