Month:  Nov 2017 Dec 2017 Jan 2018 Feb 2018 Mar 2018 Apr 2018 May 2018 Jun 2018 Jul 2018 Aug 2018 Sep 2018 Oct 2018 Nov 2018 Dec 2018 Jan 2019 Feb 2019 Mar 2019 Apr 2019 May 2019 Jun 2019 Jul 2019 Aug 2019 Sep 2019 Oct 2019 Nov 2019 Week:  Apr 9 - Apr 13 Apr 16 - Apr 20 Apr 23 - Apr 27 Apr 30 - May 4 May 7 - May 11 May 14 - May 18 May 21 - May 25 May 28 - Jun 1 Jun 4 - Jun 8 Jun 11 - Jun 15 Jun 18 - Jun 22 Jun 25 - Jun 29 Jul 2 - Jul 6 Jul 9 - Jul 13 Jul 16 - Jul 20 Year:  2016 2017 2018 2019 2020 2021 2022 2023 Login
Wednesday, May 16, 2018
Public Access

Category:
Category: All

16
May 2018
Mon Tue Wed Thu Fri Sat Sun
1 2 3 4 5 6
7 8 9 10 11 12 13
14 15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 31
8:00am [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.)