College of Liberal Arts and Sciences

# Department of Mathematics

## Algebra Seminar

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Title: Markov Number Ordering Conjectures
Speaker: Michelle Rabideau (University of Connecticut)
Time: Wednesday, September 27, 2017 at 11:15 am
Place: MONT 214Abstract: 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

Organizer: Liang Xiao