# Topology

## 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.

## The Grothendieck ring of varieties, and stabilization in the algebro-geometric setting - 2 of 2

A central theme of this workshop is the fact that arithmetic and topological structures become best behaved “in the limit”. The Grothendieck ring of varieties (or stacks) gives an algebro-geometric means of discovering, proving, or suggesting such phenomena.

In the first lecture of this minicourse, Ravi Vakil will introduce the ring, and describe how it can be used to prove or suggest such stabilization in several settings.

In the second lecture of the minicourse, Aaron Landesman will use these ideas to describe a stability of the space of low degree covers (up to degree 5) of the projective line (joint work with Vakil and Wood). The results are cognate to Bhargava’s number field counts, the philosophy of Ellenberg-Venkatesh-Westerland, and Anand Patel’s fever dream.

This is the second lecture in a two part series: part 1.

## Conjectures, heuristics, and theorems in arithmetic statistics - 2 of 2

We will begin by surveying some conjectures and heuristics in arithmetic statistics, most relating to asymptotic questions for number fields and elliptic curves. We will then focus on one method that has been successful, especially in recent years, in studying some of these problems: a combination of explicit constructions of moduli spaces, geometry-of-numbers techniques, and analytic number theory.

This is the second lecture of two: first lecture.

## Stable cohomology of complements of discriminants

The discriminant of a space of functions is the closed subset consisting of the functions which are singular in some sense. For instance, for complex polynomials in one variable the discriminant is the locus of polynomials with multiple roots. In this special case, it is known by work of Arnol'd that the cohomology of the complement of the discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered. A more general set-up is to consider the space of non-singular sections of a very ample line bundle on a fixed non-singular variety. In this case, Vakil and Wood proved a stabilization behaviour for the class of complements of discriminants in the Grothendieck group of varieties. In this talk, I will discuss a topological approach for obtaining the cohomological counterpart of Vakil and Wood's result and describe stable cohomology explicitly for the space of complex homogeneous polynomials in a fixed number of variables and for spaces of smooth divisors on an algebraic curve.