Consider the following three properties of a general group G:
Algebra: G is abelian and torsion-free.
Analysis: G is a metric space that admits a ‘norm’, namely, a translation-invariant metric d( . , . ) satisfying: d(1,gn) = |n| d(1,g) for all g ∈ G and integers n.
Geometry: G admits a length function with ‘saturated’ subadditivity for equal arguments: l(g2) = 2 l(g) for all g ∈ G.
While these properties may a priori seem different, in fact they turn out to be equivalent (and also to G being isometrically and additively embedded in a Banach space, hence inheriting its norm). The non-trivial implication amounts to saying that there does not exist a non-abelian group with a ‘norm’. We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blog post of Terence Tao.
(Joint – as D.H.J. PolyMath – with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
