Low-lying, fundamental, reciprocal geodesics

Joint IAS/Princeton University Number Theory Seminar
Topic:Low-lying, fundamental, reciprocal geodesics
Speaker:Alex Kontorovich
Affiliation:Rutgers University; Member, School of Mathematics
Date:Thursday, March 24
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/puias/2016/0324-Kontorovich

Markoff numbers give rise to extremely low-lying reciprocal geodesics on the modular surface, but it is unknown whether infinitely many of these are fundamental, that is, the corresponding binary quadratic form has fundamental discriminant. In joint work with Jean Bourgain, we unconditionally produce infinitely many low-lying (though not "extremely" so), reciprocal geodesics on the modular surface, settling a question of Einsiedler-Lindenstrauss-Michel-Venkatesh. Some ingredients are expander graphs, sieving on thin orbits, and producing a level of distribution "beyond expansion". No prior knowledge of these topics will be assumed.