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? 

Cannon-Thurston maps

Symplectic quotient singularities

If a finite group G acts on a complex vector space V then the set of orbits V/G is naturally an affine variety. It is almost always singular. Beginning with my PhD work, I’ve always been interested in resolutions of these singular spaces. I’ll explain what a “symplectic resolution” is, when these exist and why you might care about them. The main point of the talk is to use this example to illustrate some of the things that you’ll typically encounter, or require, in your research career e.g., luck, bad luck, hard work, mistakes… In other words, I’ll use this example to tell you some of the things that have happened to me during the almost 20 years spent thinking about the problem. 

Feynman Graph Theory

Feynman graphs (or diagrams) and their associated Feynman Integrals are the building blocks of perturbative Quantum Field Theory, and thus serve as a basis for Standard model theory predictions for experiments such as the LHC. The higher the precision required the more complicated the integrals become. But their mathematical properties are also interesting in their own right. I will discuss the Hopf algebras of Feynman graphs, which underly renormalisation of UV and certain IR divergences, as well as certain asymptotic expansions. I will conclude with some applications in collider physics and condensed matter physics.

Re-innovating the wheel: Tannaka duality and extended operators in QFT

Tannaka duality was developed in the late 1930’s in order to reconstruct a complex Lie group from its category of representations. Its generalization, more often simply called “reconstruction theory,” has now become a fundamental part of representation theory — in particular, representation theory of categories and higher categories. It was used in physics in the 1990’s (and the 2000’s… and again in the 2010’s…) to represent line operators in topological QFT’s — by folks like Freed, Witten, and Costello. I will talk about how I stumbled into it in the course of my work, was drawn in by its power, and have been seeking new applications.

Symmetries of Renormalization Group Fixed Points

I will give a general overview of our current understanding of the symmetries of renormalization group fixed points in quantum field theory and why they are interesting, including scale, conformal and Weyl invariance. I will describe the distinction between these symmetries, as well as what is generally known about when these symmetries imply one another. I’ll also give an overview of my past and present work on this topic and how it’s evolved over my career.

Before and after the cobordism hypothesis

In 1995, James Dolan and I formulated a series of “hypotheses” about topology and higher categories, before the theory of higher categories was sufficiently developed to make these into precisely stated conjectures. In 2009 Jacob Lurie reformulated one of these using infinity-categories, called it the “cobordism hypothesis”, and gave a 111-page outline of a proof. Simply put, the cobordism hypothesis gives a purely algebraic description of smooth manifolds, helpful for constructing topological quantum field theories. I’ll say a bit about what led Dolan and me to this hypothesis, what it says, why it’s useful, and why I quit working on it.

Can you hear the shape of a drum?