Webinar on A Crowd-Sourced, Computer-Assisted Analysis-Definition of Abelian Groups

Organizer: Math Ecstasy

About the Session

• When: Saturday, August 15, 2026
• Time: 7:00 PM – 8:30 PM (IST)
• Online (Google Meet)
• Free Registration

N.B.: There won't be any certificate of participation

Speaker: Prof. Apoorva Khare, Indian Institute of Science, Bangalore

Organizing Team

Dr. Vinod Kumar P. – Associate Professor of Mathematics, T. M. Govt. College, Tirur, Kerala

Dr. Bijumon R. – Professor of Mathematics, MG College, Iritty, Kerala

Dr. Rosna Paul – Research Assistant (Postdoc), Fern University in Hagen, Germany

Ms. Gayathri M. – Research Fellow, IMSc, Chennai

Ms. Amritha Varrier – Research Fellow, IIT Madras

Ms. Ardra A. N. – Research Fellow, University of Hyderabad

Ms. Arya E. K. – Research Fellow, IISER Thiruvananthapuram

Ms. Tithi Biswas – Research Fellow, IIT Ropar

Moderator: Ms. Ajuna N. P.

Abstract of the Talk

Consider the following three properties of an arbitrary 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", and this talk aims to present the proof, some motivations and connections, and time permitting, the logistics of how the problem was solved via a PolyMath project that began on a blogpost of Terence Tao. (Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)