Évènement

Matthew Litman
University College, Dublin
https://people.ucd.ie/matthew.litman

Date(s) : 06/05/2025   iCal
11h00 - 12h00

Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p, whereas it was conjectured in 1991 by Baragar that these graphs are connected for all p.

In this talk, we discuss our recent work confirming this conjecture for all primes p>3448×10³⁹² by employing results of Chen and Bourgain, Gamburd, and Sarnak. We introduce the notion of maximal divisors as a key tool in our proof and prove sharp asymptotic and explicit upper bounds on the number of them. After showing connectivity in the standard Markoff setting, we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P¹xP¹xP¹ cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of BGS, we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily Wₖ of such surfaces, we construct finite orbits in Wₖ(C) by studying small orbits that appear in Wₖ(Fₚ) for many values of p and k. This shows that the connectivity property for the standard Markoff case fails for its projective counterpart.

This talk is based on separate joint works with J. Eddy, E. Fuchs, D. Martin, and N. Tripeny, as well as with E. Fuchs, J. Silverman, and A. Tran respectively.

Emplacement
I2M Luminy - TPR2, Salle 210-212 (2e étage)

Catégories

• Séminaire