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: $d(1,g^n) = |n| d(1,g)$ for all $g \in G$ and integers $n$.
Geometry: $G$ admits a length function with “saturated” subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g \in G$.
While these properties may a priori seem different, in fact they turn out to be equivalent. The nontrivial 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 if time permits, the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)