Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

The Markoff equation is the Diophantine equation $x^2 + y^2 + z^2 - 3xyz = 0$. It's solutions are not so hard to enumerate; they form a tree, rooted at $(1, 1, 1)$, with edges given by the so-called Vieta involution. Less understood are the solutions to the Markoff equation modulo a prime $p$. Modulo $p$, the solutions form graphs $X^*(p)$, and the shape of these graphs is the subject of several open questions. It is conjectured that $X^*(p)$ is connected for every prime $p$. This conjecture being true would have strong implications for number-theoretic approaches to answering questions about the Markoff solutions in general. In this talk, we summarize the current state of this conjecture, discuss the current approaches to deciding if $X^*(p)$ is connected for various primes $p$, and preview possible avenues for proving connectivity for the values of $p$ for which we do not yet know about the connectivity of $X^*(p)$.

Pizza at 11:50am