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[4:00pm] Mathematics Colloquium
 Description:
 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 torsionfree.
(2) Analysis: G is a metric space that admits a "norm", namely, a
translationinvariant 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 nonabelian 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/biinvariantmetricsoflineargrowthonthefreegroup/
of Terence Tao.
(Joint  as D.H.J. PolyMath  with Tobias Fritz, Siddhartha Gadgil, Pace
Nielsen, Lior Silberman, and Terence Tao.)


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