Consider the following three properties of an arbitrary group G:

(1) Algebra: G is abelian and torsion-free.

(2) 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.

(3) Geometry: G admits a length function with “saturated” subadditivity for equal arguments: l(g^2) = 2l(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 finally, 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.)

About the speaker: 

Apoorva Khare is Associate Professor of Mathematics at the Indian Institute of Science (IISc), Bangalore. After receiving his BStat from the Indian Statistical Institute (Kolkata) and MS+PhD from the University of Chicago, he worked at the University of California Riverside, Yale, and Stanford before joining IISc as a faculty member. Apoorva's interests include positivity and analysis (he authored a 2022 book with Cambridge University Press and TRIM), combinatorics and discrete mathematics, and representation theory. He also introduced in 2011 a Quantitative Reasoning course for non-math majors at Yale, and coauthored a textbook for it through Yale University Press in 2015.