Topology

Hurwitz numbers are counts of maps between Riemann surfaces with specified ramification profiles. Alternatively, they may be seen as counting decompositions of the identity in symmetric groups into permutations of given cycle type or as certain expressions of symmetric functions. While these two interpretations, due to Hurwitz, Frobenius, and Schur, have been known for over a hundred years, these numbers occur in more contexts: they give solutions to certain systems of PDEs, such as the Kadomtsev-Petviashvili hierarchy, they encode intersection numbers of moduli spaces of curves, and they can be found via Eynard-Orantin topological recursion.

In this talk, I will first give some of the definitions of Hurwitz numbers and then explain what topological recursion is and how it helps us shed new light on these numbers.

Speaker Biography

Reinier Kramer studied physics and mathematics at the Universities of Amsterdam and Cambridge. In 2019, he obtained a PhD at the University of Amsterdam with Sergey Shadrin, and from 2019 to 2021 he held a postdoctoral fellowship at the Max Planck Institute of Mathematics in Bonn, in the group of Gaëtan Borot. He is currently a postdoctoral fellow with Vincent Bouchard at the University of Alberta. He works in the areas of mathematical physics and algebraic geometry, and is mainly interested in using topological recursion to calculate intersection-theoretic and enumerative-geometric objects, with a focus on Hurwitz numbers.

yperbolic Lattices are tessellations of the hyperbolic plane
using, for instance, heptagons or octagons. They are relevant for quantum
error correcting codes and experimental simulations of quantum physics in
curved space. Underneath their perplexing beauty lies a hidden and,
perhaps, unexpected periodicity that allows us to identify the unit cell
and Bravais lattice for a given hyperbolic lattice. This paves the way for
applying powerful concepts from solid state physics and, potentially,
finding a generalization of Bloch's theorem to hyperbolic lattices. In my
talk, I will explain some of the mathematics underlying this hyperbolic
crystallography.

For other events in this series see the quanTA events website
.

Transfer systems are discrete objects that encode the homotopy theory of N∞ operads, i.e., the operads whose algebras are homotopy commutative monoids with a class of equivariant transfer (or norm) maps. They have a rich combinatorial structure defined in terms of the subgroup lattice of the group of equivariance, G. Indeed, if G is a cyclic p-group, there are Catalan-many transfer systems that assemble into the Tamari lattice (i.e., associahedron). In this talk, I will show that when G is finite Abelian, transfer systems are in natural bijection with weak factorization systems on the poset category of subgroups of G. This leads to a novel involution on the lattice of transfer systems, generalizing an observation of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. I will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.

This is joint work with Angélica Osorno and a team of Reed College undergraduates: Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh (the Electronic Collaborative Mathematics Research Group, or eCMRG).

Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g., K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss ongoing work with Keegan Boyle trying to understand the equivariant 4-genus.

A triangulation of a surface is a way to divide it into a finite number of triangles. Let us pick a random triangulation uniformly among all those with a fixed size and genus. What can be said about the behaviour of these random geometric objects when the size gets large? We will investigate three different regimes: the planar case, the regime where the genus is not constrained, and the one where the genus is proportional to the size. Based on joint works with Baptiste Louf, Nicolas Curien and Bram Petri.