I gave a couple of lectures on q-analogs in enumerative combinatorics. The first two are from a course I took in spring 2023. The last one was from a directed reading in fall 2022.

Introduction to q-analogs

[Notes] q-analogs generalize combinatorial objects into functions of a formal variable q (and recover the original concept when we send q to 1). We prove that the q-analog for n! provides us with the generating function for inversions, and we explore basic results regarding the q-binomial coefficient. With time permitting, we discuss where q-analogs come up in the study of symmetric functions

Using q-hypergeometric series to prove Euler’s Pentagonal Number Theorem

[Notes] Euler’s pentagonal number theorem establishes a beautiful picture of the partition generating function. q-hypergeometric series are a hammer of modern enumerative combinatorics which makes the proof of this theorem quite easy.

Statistical Mechanics Helps Us Count Alternating Sign Matrices

[Notes] The enumeration of alternating sign matrices (ASMs) presents a profound, deceptively simple combinatorial problem, with wide-ranging connections across mathematics. These objects have a particularly interesting correspondence with statistical mechanics and, in particular, a lattice model called “square ice.” In this presentation, we motivate the study of ASMs and sketch Greg Kuperberg’s 1995 proof of the ASM Conjecture, which uses prominent tools of statistical mechanics such as the Yang-Baxter Equation.