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### Navigating the Markoff graph computationally

**Algebraic Geometry and Number Theory**

Speaker: | Colby Brown, UC Davis |

Related Webpage: | https://math.ucdavis.edu/~colbyabrown/ |

Location: | 2112 MSB |

Start time: | Tue, Oct 15 2024, 1:10PM |

The Markoff graphs modulo p were proven by Chen (2024) to be connected for all but finitely many primes, and Baragar (1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation modulo p lifts to a solution over Z. In this talk, we provide an algorithmic realization of the process introduced by Bourgain, Gamburd, and Sarnak to test whether the Markoff graph modulo p is connected for arbitrary primes. Our algorithm runs in $o(p^{1+\varepsilon})$ time for every $\varepsilon$ > 0. We demonstrate this algorithm by confirming that the Markoff graph modulo p is connected for all primes less than one million. Finally, we discuss other approaches to decomposing the Markoff graph, with possible applications to determining its spectral properties.