Title: Fundamental Groups of Random Knots
Abstract: 
We seek statistical regularity in the etale fundamental groups of Spec Z punctured at a growing  prime number p (the Galois group of the maximal extension of the rational numbers unramified away from p). 
As a first step, following the paradigm of arithmetic topology, we show that the (profinite completion of the) fundamental group of a random knot is distributed according to a certain measure. The proof utilizes the Sawin—Wood machinery for constructing a measure from moments.
This result is then used to show that the very same measure governs the distribution of etale fundamental group of the affine line over a finite field punctured at growing irreducible polynomial P (in a suitable asymptomatic regime).
The talk is based on a joint work in progress with Guy Blachar.
Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=c747a421-80b0...