Dynamical thresholds in high dimensional landscapes

I will survey recent progress on understanding dynamical thresholds in two problems: spherical spin glasses and spiked tensor models. I will begin by reviewing an elementary approach to studying spherical spin glass dynamics based on differential inequalities for one-time observables. Using this approach, one can obtain an approximate phase diagram for the evolution of the energy H and its gradient under Langevin dynamics. I will then turn to how these ideas can be used to understand the algorithmic thresholds in spiked tensor models discussed in Monday's talk. In particular, I will discuss the key ideas behind the proof of these thresholds by combining global regularity estimates for the landscape with point-wise estimates for the initialization. This is joint work with Ben Arous-Gheissari.