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Wednesday, May 16, 2018
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4:00pm [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 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.)

5:00pm