Event
Michelle Rabideau, University of Connecticut
Friday, October 27, 2017 13:30to14:30
Room PK-4323, Seminar LACIM, 201 Ave. President-Kennedy, CA
Markov number ordering conjectures.
A Markov number is a number in the triple $(x,y,z)$ of positive integer solutions to the Diophantine equation $x^2+y^2+z^2 = 3xyz$. Markov numbers are a classical topic in number theory related to many areas of mathematics such as combinatorics and cluster algebras. Markov numbers are related to cluster algebras by Markov snake graphs, where a Markov snake graph is the snake graph of a cluster variable of the once punctured torus. The number of perfect matchings of a Markov snake graph, given by the numerator of the associated continued fraction, is a Markov number. In this talk, we discuss three conjectures given in Martin Aigner’s book [A] that provide an ordering on the Markov numbers $m_{p/q}$ for a fixed numerator $p$, fixed denominator $q$ and a fixed sum $p+q$. [A] M. Aigner, Markov's theorem and 100 years of the uniqueness conjecture, Springer 2010.