Varieties with Non-Commuting Involutions and Connectivity of the Graph of Mod p Points

Pre-Seminar Title: Fun and Games with the Markoff Equation
Abstract pre-seminar: It is an amazing fact that all of the non-zero integer solutions to the Markoff equation x^2+y^2+z^2=3xyz can be derived starting from the basic solution (1,1,1). I'll explain how this is done, describe the infamous "Unicity Conjecture", and as time permits, look at similar results for a wider class of surfaces.

Seminar Title: Varieties with Non-Commuting Involutions and Connectivity of the Graph of Mod p Points
Abstract: The Markoff equation M : x^2+y^2+z^2=3xyz is an example of a surface having three non-commuting involutions, which Markoff used to give a complete description of the integer solutions. A natural local-global question is whether every solution over the finite field F_p lifts to a solution over Z, or equivalently, whether the function graph of the non-zero F_p points is connected. A conjecture of Baragar (1990s) says that the answer is yes, and work of Bourgain-Gamburd-Sarnak (2015-2025) and W. Chen (2023) resolves the question for all sufficiently large p.  In this talk I will describe their work, and then I will discuss joint work with Fuchs, Litman, and Tran studying  analogous problems for K3 surfaces X defined by (2,2,2)-forms in P^1xP^1xP^1 that similarly have three non-commuting involutions.