# Topology

## 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.

## 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.

## The circle method and the cohomology of moduli spaces of rational curves

The cohomology of the space of degree d holomorphic maps from the complex projective line to a sufficiently nice algebraic variety is expected to stabilize as d goes to infinity. The limit is expected to be the cohomology of the double loop space, i.e. the space of degree d continuous maps from the sphere to that variety. This was shown for projective space by Segal, and there has been further subsequent work. In joint work with Tim Browning, we give a new approach to the problem for smooth affine hypersurfaces of low degree (which should also work for projective hypersurfaces, complete intersections, and/or higher genus curves), based on methods from analytic number theory. We take an argument of Birch that solves the number-theoretic analogue of this problem and translate it, step by step, into the language of ell-adic sheaf theory using the sheaf-function dictionary. This produces a spectral sequence that computes the cohomology, whose degeneration would imply that the rational compactly-supported cohomology matches that of the double loop space.

## $E_2$ algebras and homology - 2 of 2

Block sum of matrices define a group homomorphism $GL_n(R) \times GL_m(R) \to GL_{n+m}(R)$, which can be used to make the direct sum of $H_s(BGL_t(R);k)$ over all $s, t$ into a bigraded-commutative ring. A similar product may be defined on homology of mapping class groups of surfaces with one boundary component, as well as in many other examples of interest. These products have manifestations on various levels, for example there is a product on the level of spaces making the disjoint union of $BGL_n(R)$ into a homotopy commutative topological monoid. I will discuss how it, and other concrete examples, may be built by iterated cell attachments in the category of topological monoids, or better yet $E_2$ algebras, and what may be learned by this viewpoint. This is all joint work with Alexander Kupers and Oscar Randal-Williams.

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

## Coincidences between homological densities, predicted by arithmetic - 2 of 2

In this talk I'll describe some remarkable coincidences in topology that were found only by applying Weil's (number field)/(f unction field) analogy to some classical density theorems in analytic number theory, and then computing directly. Unlike the finite field case, here we have only analogy; the mechanism behind the coincidences remains a mystery. As a teaser: it seems that under this analogy the (inverse of the) Riemann zeta function at $(n+1)$ corresponds to the 2-fold loop space of $P^n$. This is joint work with Jesse Wolfson and Melanie Wood.

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

## Representation stability and asymptotic stability of factorization statistics

In this talk we will consider some families of varieties with actions of certain finite reflection groups – such as hyperplane complements or complex flag manifolds associated to these groups. The cohomology groups of these families stabilize in a precise representation theoretic sense. Our goal is to explain how these stability patterns manifest, and can be recovered from, as asymptotic stability of factorization statistics of related varieties defined over finite fields.

## $A^1$ enumerative geometry: counts of rational curves in $P^2$ - 2 of 2

We will introduce $A^1$ homotopy theory, focusing on the $A^1$ degree of Morel. We then use this theory to extend classical counts of algebraic-geometric objects defined over the complex numbers to other fields. The resulting counts are valued in the Grothendieck--Witt group of bilinear forms, and weight objects using certain arithmetic and geometric properties. We will focus on an enrichment of the count of degree d rational plane curves, which is joint work with Jesse Kass, Marc Levine, and Jake Solomon.