Gérard Ben Arous - New York University What makes a high dimensional optimization problem hard? Is it the topological complexity of its landscape: many critical points, topologically complex level sets? or is it the weakness of the signal in the regions of high entropy? In other words, is it hard to escape mediocrity or hard to avoid traps? Or both? So, when are the usual local algorithms (GD or SGD) provably good? I will discuss some recent advances using a very useful workhorse (Tensor PCA), and introduce some preliminary understanding on other models. This talk will be based on recent work with Reza Ghessairi and Aukosh Jagannath, and with Giulio Biroli and Antoine Maillard.