# Topology

## Hurwitz Numbers via Topological Recursion

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.

## An Overview of Knots and Gauge Theory

The Jones polynomial of a knot, discovered in 1983, is a very

subtle invariant that is related to a great deal of mathematics and

physics. This talk will be an overview of quantum field theories in

dimensions 2, 3, 4 and 5 that are intimately related to the Jones

polynomial of a knot and a more contemporary refinement of it that is known

as Khovanov homology.

## Crystallography of hyperbolic lattices: from children's drawings to Fuchsian groups

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

.

## Characterizing handle-ribbon knots

Kauffman conjectured that a knot K is slice if and only if it bounds a genus-g Seifert surface containing a g-component slice link as a cut system. It’s very easy to show that a knot is ribbon if and only if it bounds a genus-g Seifert surface containing a g-component unlink as a cut system. Alex Zupan and I proved something in the middle of these statements: a knot is handle-ribbon (aka strongly homotopy-ribbon, aka something I will define in the talk) if and only if it bounds a genus-g Seifert surface containing a g-component R link L as a cut system—meaning that zero-surgery on L yields #_ g S^1 × S^2 . This gives a 3-dimensional definition of a 4-dimensional property. I’ll talk about these 3.5D knot properties and maybe how we use these techniques to extend a statement of Casson and Gordon. (The work in this talk is joint with Alexander Zupan from the University of Nebraska–Lincoln.)

## Weak factorization and transfer systems

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).

## Representation stability and configurations of disks in a strip

Representation stability, formalized in 2012 by Church, Ellenberg, and Farb, is a property exhibited by the homology of the configuration space of points in the plane: even as the number of points goes to infinity, the jth homology is generated by cycles in which at most 2j of the points move. What about the configuration space of disks of width 1 in an infinite strip of width w? This disks in a strip space behaves more like the no-k-equal configuration space of the line, where k-1 but not k points may be collocated; we show that the homology of this no-k-equal space exhibits generalized representation stability as defined by Sam–Snowden and Ramos. The method is to compute homology combinatorially using discrete Morse theory. Unlike other examples of homology with generalized representation stability, here the asymptotic behavior depends on the degree of homology.

## Symmetric knots and the equivariant 4-ball genus

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.

## Enumerative geometry via the A^1-degree

Morel's $A^1$ -degree in $A^1$-homotopy theory is the analog of the Brouwer degree in classical topology. It takes values in the Grothendieck-Witt ring $GW(k)$ of a field $k$, that is the group completion of isometry classes of non-degenerate symmetric bilinear forms. We can use the $A^1$ -degree to count algebro-geometric objects in $GW(k)$, giving an $A^1$-enumerative geometry over non-algebraically closed fields. Taking the rank and the signature recovers classical counts over the complex and the real numbers, respectively. For example, the count of lines on a smooth cubic surface enriched in $GW(k)$ has rank 27 and signature 3.

## Random discrete surfaces

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.

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## The stable cohomology of the moduli space of curves with level structures

I will prove that in a stable range, the rational cohomology of the moduli space of curve with level structures is the same as the ordinary moduli space of curves: a polynomial ring in the Miller-Morita-Mumford classes.