Talk

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.

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.

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

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.

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

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,

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

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.

Time: Tuesdays, 4.30 - 6 pm
Place: MPIM Lecture Hall, Vivatsgasse 7
First lecture: on April 2, 2019, end on July 2