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.)

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.)

Description

Ramanujan Hall

Date

Wed, May 16, 2018

Start Time

4:00pm-5:00pm IST

Duration

1 hour

Priority

5-Medium

Access

Public

Created by

DEFAULT ADMINISTRATOR

Updated

Thu, May 3, 2018 11:50am IST