Title: Groups with norms: a PolyMath adventure
Speaker: Apoorva Khare (Indian Institute of Science)
Abstract:
Consider the following three properties of a general 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) = 2 l(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
http://michaelnielsen.org/polymath1/index.php?title=Linear_norm
that began on a
blogpost
https://terrytao.wordpress.com/2017/12/16/bi-invariant-metrics-of-linear-growth-on-the-free-group/
of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace
Nielsen, Lior Silberman, and Terence Tao.)