Knots Invariants and Arithmetic Statistics

Abstract: The Grothendieck school introduced étale topology to attach algebraic-topological invariants such as cohomology to varieties and schemes. Although the original motivations came from studying varieties over fields, interesting phenomena such as Artin–Verdier duality also arise when considering the spectra of integer rings in number fields and related schemes. A deep insight, due to B. Mazur, is that through the lens of étale topology, spectra of integer rings behave as $3$-dimensional manifolds while prime ideals correspond to knots in these manifolds. This knots and primes analogy provides a dictionary between knot theory and number theory, giving some surprising analogies. For example, this theory relates the linking number to the Legendre symbol and the Alexander polynomial to Iwasawa theory. In this talk, we shall start by describing some of the classical ideas in this theory. I shall then proceed by describing how via this theory, giving a random model on knots and links can be used to predict the statistical behavior of arithmetic functions. This is joint work with Ariel Davis.

number theory

Audience: researchers in the topic

Harvard number theory seminar