Apoorva Khare

Indian Institute of Science, Bangalore

Apoorva Khare

Session 1B: Inaugural Lectures by Fellows / Associates

Polymath 14: A crowd-sourced, computer-aided analysis-definition of abelian groups

Abstract: We answer a basic question: What are all the groups that have a norm?

This project connects across algebra, analysis, geometry, probability, and combinatorics. It also provides a nonstandard, modern model for mathematics collaboration in today's fast-evolving world, using both technology (in multiple crucial ways) and crowdsourcing. Research was conducted round-the-clock across multiple timezones and continents, by six colleagues (with contributions from several others), and progress was tabulated on the blog of Terence Tao (UCLA), including via help from a computer. The findings reveal a new identification: a fundamental algebraic structure (abelian torsionfree) is precisely the same as a fundamental analysis structure (norm), for an arbitrary group.