Markoff Numbers and geodesic lengths

Greg McShane (Institut Fourier)

Monday 7th June 16:00-17:00 Online

Abstract

Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic x2 y2 z2 xyz.

A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras.

In the 50's, H. Cohn discovered a relationship between Markov numbers and the lengths of closed simple geodesics on the punctured torus. In the 90's, Igor Rivin and myself introduced a method which allows one to give estimates for the number of Markov numbers less than L>0. The key ingredient in this was the use of a norm on the first homology.
 
We will :
- explain the geometry of the norm and how it can be used to prove new identities for lengths of simple closed geodesics
- use the convexity to prove a new unified proof of certain conjectures from Martin Aigner's book, Markov's Theorem and 100 Years of the Uniqueness Conjecture.

 

The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/91412568415 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).


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