Mathieu Guay-Paquet completed his PhD in 2012 under the supervision of Ian Goulden. His thesis, entitled "Algebraic Methods and Monotone Hurwitz Numbers", was examined by Fields medallist Andrei Okounkov.
Mathieu's doctoral thesis deals with a new object, the monotone Hurwitz number, which counts the number of ways of expressing a specified permutation as a product of a given number of transpositions, subject to technical conditions on "transitivity" and "monotonicity". Without the monotonicity restriction, these numbers are called Hurwitz numbers, and have been extensively studied in mathematical physics, algebraic geometry, and algebraic combinatorics since Witten made his famous conjecture about twenty years ago. Witten's conjecture, soon proved by Kontsevich, then by Okounkov and Pandharipande, and a number of other authors, established a connection between two-dimensional gravity, enumerative geometry, and integrable hierarchies. In his thesis, Mathieu developed new algebraic methods that allowed him to prove a number of difficult results for monotone Hurwitz numbers and their generating series. These results have remarkable parallels to the known results for Hurwitz numbers. The strength of these parallels was quite unexpected, and seems highly highly mysterious even after-the-fact.
monotone Hurwitz numbers and integral hierarchies. The most basic of
these is that particular generating series for monotone Hurwitz numbers
satisfy the Kadomtsev-Petriashvili (KP) hierarchy.
Since completing his PhD, Mathieu has joined Université du Québec à Montréal (UQAM) as holder of an Natural Sciences and Engineering Research Council of Canada (NSERC) Postdoctoral Fellowship.