Laura DeMarco (Harvard)

About 20 years ago, Shouwu Zhang formulated the “Dynamical Manin-Mumford” conjecture about the geometry of (pre-)periodic points for endomorphisms of projective varieties; it remains open today.  The conjecture generalized the well-known Manin-Mumford Conjecture (theorem of Raynaud, 1983) about torsion points in abelian varieties.  Proofs of all known cases use arithmetic tools, but very little is known beyond the setting of abelian varieties. I will discuss this conjecture and related questions, emphasizing the number-theoretic ingredients we have used.