Video Lectures

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Two or infinity

Dan Cristofaro-Gardiner

It is conjectured that every Reeb flow on a closed three-manifold has either two, or infinitely many, simple periodic orbits. I will survey what is currently known about this conjecture. Then, I will try to explain some of the key ideas behind...

(Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties Ag for any value of g. For g<16 the dimension is g−1, while for g≥16, it is determined by the larged dimensional compact Shimura subvariety, which we determine. Our methods rely on deforming the boundary using special varieties, and functional transcendence theory.