Numerical polynomials, Kummer congruences and Massey products

Andrew Baker (University of Glasgow)

Wednesday 13th October 16:00-17:00 Maths 110

Abstract

This talk will involve some Hopf algebras, some number theory and some homological algebra. It is meant to be accessible to a broad audience.

Rings of numerical polynomials crop up in many places including numerical analysis, algebraic combinatorics in the spirit of G-C. Rota et al, number theory and algebraic topology. I will set the scene by introducing some rings of numerical polynomials for the integers and their Hopf algebra structures. Then I will explain connections with $p$-adic integration and in particular a way to encode the Kummer congruences involving Bernoulli numbers. Finally I will show how this can be used to determine certain Massey products that arise in the cohomology of the Hopf algebra of stably numerical polynomials, a topic of interest in stable homotopy theory.

[This is an overview of work over several decades with many people, especially Francis Clarke]

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