Theo Douvropoulos: Factorizations in symmetric and Coxeter groups: combinatorial and geometric perspectives

Theo Douvropoulos

In how many "different" ways can you draw a graph?
How many (degree-n, monic) polynomials exist with a fixed set of critical values?

The answers to both questions are related to enumerating certain classes of factorizations in the symmetric group (different ones in each case). The first leads to the notion of a "combinatorial map", developed extensively the last 50 years particularly in France, and the second goes back to Hurwitz in the late 1800s who first observed that a natural monodromy construction associated to branched coverings of the sphere is encoded precisely via factorizations. Both directions have grown rapidly in the last few decades: New discrete structures have emerged, and the theory has been extended to Coxeter groups, both combinatorially and through singularity theory; new, refined geometric connections have been established to the moduli space of stable curves.

I will present this story, which forms the context for one of my main research programs. I will elaborate on connections to other fields and some of my contributions, and discuss one of my most exciting new research projects that connects the (algebraic) geometry of monodromy factorizations to the (discrete) geometry of certain polytopes.