On some conjectures and questions of Talagrand on positive selector processes and empirical processes
Understanding suprema of stochastic (empirical) processes is an important subject in probability theory with many applications. In the Gaussian case, via generic chaining and Talagrand's celebrated majorizing measure theorem, Talagrand showed that extreme events of suprema of Gaussian processes admit simple geometric characterization. Moving beyond the Gaussian case is a much more challenging quest. In this talk, I will discuss recent joint work with Jinyoung Park that resolves a conjecture of Talagrand on extreme events of suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on general positive empirical processes. Combining with the recent resolution of the (generalized) Bernoulli conjecture and advances in chaining, this gives the first steps towards the last missing piece in the study of suprema of general empirical processes. I will also touch on how one of the ideas in our proof of Talagrand's conjecture leads to the proof of the Kahn–Kalai conjecture, an important question in probabilistic combinatorics and random graph theory.