Uncounting polynomials

For my research day job, I solve differential equations on computers. I do this by appropriating many of the special functions of mathematical physics to keep track of all kinds of information in numerical form. But looking at the inner workings of those functions leads to the striking observation: spherical harmonics, Bessel Functions, Chebyshev polynomials, Hermite functions, Bernoulli polynomials, and many more are all littered with the trappings of combinatorics: factorials, monomials, binomial coefficients, Stirling numbers, and much more.

Often, combinatorics researchers apply their trade by proposing a concrete or abstract situation and hunt for formulas that “count” the number of possibilities. The process sometimes works in reverse, but much less so. In that case, you start with a formula and ask, “What real-life scenario is this counting?” However, a perennial difficulty about “uncounting” orthogonal polynomials is the common appearance of fractions, and even worse, negative numbers! To be clear: I want something that, when properly uncounted, could be performed on stage at the Fringe Festival. What are negative actors, exactly?

In this talk, I’ll outline some of the progress my PhD student, Miru Park, and I have been making on uncounting Laguerre polynomials and related problems. In the process, we’ve discovered that modern Category Theory seems to contain just the right tools: groupoids, weak actions, and the Grothendieck group for those pesky negatives. I’ll do my best to explain everything as elementally as possible, with the ultimate goal of using the abstract to uncover the concrete.