## Malle's Conjecture for octic $D_4$-fields

We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant.

## The generator-relator models for class groups of number fields

We will discuss a generator-relator random group model that provides probability measures agreeing with the Cohen-Lenstra heuristics and the Cohen-Martinet heuristics. We will also discuss the “good primes” in our random group model. This is a joint work with Melanie Matchett Wood.

## Justifying Random Models of Nonabelian Class Groups

Recent works by Boston, Wood, and others suggest a balanced presentation random model for 'nonabelian class groups'. We show that over function fields, these class groups (etale fundamental groups) are indeed balanced.

## Large gaps between prime divisors

It is known that the set of prime factors of integers behaves more or less like a Poisson point process. We explain several properties proved in Smith's work regarding prime divisors, in particular that almost all squarefree integers have a certain form of large gap between prime divisors.

## Arithmetic statistics via graded Lie algebras

I will talk about recent work with Jack Thorne in which we find the average size of the 3-Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

## Families of elliptic curves ordered by conductor

Conjectures on the statistics of elliptic curves are generally formulated with the assumption that the curves in question are ordered by their conductors. However, when proving results on the

## Redei's reciprocity law

I explain a corrected 21st century reformulation and give the first complete proof of a reciprocity law from 1939 that is due to Redei, and that played a central role in Smith's 2016 preprint on quadratic 2-class groups.

## Spins, governing fields and negative Pell

Let K be a cyclic, totally real extension of Q of degree at least 3 and let sigma be a generator of Gal(K/Q). We further assume that the totally positive units are exactly the squares of units. In this case, Friedlander, Iwaniec, Mazur and Rubin define the spin of an odd principal ideal to be spin(sigma, a) = (alpha/sigma(alpha))_K, where alpha is a totally positive generator of a and (*/*)_K is the quadratic residue symbol in K. They then proceed to prove equidistribution of spin(sigma, p) as p varies over the odd principal prime ideals of K.

## Negative Pell, Higher Redei reciprocity, Higher genus theory and Arakelov ray class groups

n this talk we shall consider precise instances of the following question. Let S be a finite set of rational places, how does the unit group of a random real quadratic field typically embed in the various completions above S? An instance of this question, when S consists of the infinite place, is given by Stevenhagen's conjecture. I will explain a joint work with P. Koymans, where we make progress on this conjecture by means of Smith's method. I will explain our two main innovations. The first one is a generalization of Redei's reciprocity to governing expansions.

## Distributions of unramified extensions of global fields

We give conjectures on the distribution of the Galois groups of the maximal unramified extensions of Galois \Gamma number fields or function fields for any finite group \Gamma (for the part of the

## A Primer on Counting Solutions to Central Embedding Problems

Alex Smith recently verified the distribution of the $2^\infty$-part of the class group of imaginary quadratic fields predicted by Gerth, Cohen, and Lenstra. Despite the fact that

## Geometric arithmetic statistics

We will explain the general philosophy by which one can relate questions in arithmetic statistics over Q to analogous questions over function fields of curves over F_q, and from there to questions about the topology of moduli spaces of complex algebraic varieties and morphisms. Our primary example will be the result of Ellenberg-Venkatesh-Westerland on geometric Cohen-Lenstra and the cohomology of Hurwitz spaces, but we will also try to indicate what this program might look like in the context of variation of Selmer groups,

## Distributions of unramified 2-group extensions of quadratic fields

This is a continuation of the talk by Brandon Alberts. We will discuss the Cohen-Lenstra heuristics from the point of view of counting unramified number field extensions of quadratic fields. We will apply the field counting results discussed in the previous talk to prove distributional results.

## $2$-parts of class groups in thin families

We will give an overview of what is known about the statistics of $2$-parts of class groups of quadratic number fields parametrized by discriminants of the form $dp$, where $d$ is fixed and $p$ varies over the set of prime numbers. In these thin families, the map sending a prime $p$ to the $2^k$-rank of the class group of $\mathbb{Q}(\sqrt{dp})$ is Frobenian for $k\leq 3$ (a result of Peter Stevenhagen) and likely not Frobenian for $k\geq 4$. We will summarize

## Class groups, Selmer groups and Cassels--Tate pairings

In this largely expository talk we will give an introduction to certain Selmer groups associated to (compatible systems of) finite Galois modules, and describe an analogue of the Cassels--Tate pairing in this setting, due to Flach. We will then discuss the effect of twisting these modules by quadratic characters and show how class groups of quadratic fields, and Selmer groups of quadratic twist families of elliptic curves, naturally fit into this framework. If time permits we will discuss additional objects of arithmetic interest which can be understood from this viewpoint.

## Recent developments in Quantum Topology

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

## Gromov-Witten theory with expansions

The subject of this talk will be the geometry of the moduli space of stable maps, relative to a normal crossings divisor, and a degeneration formula for the resulting Gromov-Witten theory. The moduli space and the formula together generalize a well-known package, developed by Jun Li, for smooth divisors. The main ingredient is a virtual weak semistable reduction theorem, controlled by the polyhedral geometry of tropical curves and their moduli. I will give an introduction to this circle of ideas, focusing on the very simple case of rational curves in Pn.

## Vertex Operator Algebras and Modular Forms

Time: Tuesdays, 4.30 - 6 pm

Place: MPIM Lecture Hall, Vivatsgasse 7

First lecture: on April 2, 2019, end on July 2

## Recent developments in Quantum Topology -- Cancelled --

We will review the basics of quantum topology such as the colored Jones polynomial of a knot, its standard conjectures relating to asymptotics, arithmeticity and modularity, as well as the recent quantum hyperbolic invariants of Kashaev et al, their state-integrals and their structural properties. The course is aimed to be accessible by graduate students and young researchers.

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