Huy Tuan Pham (Stanford) - Talagrand’s selector process conjecture and suprema of positive empirical processes

Understanding suprema of stochastic processes is an important subject in probability theory with many applications. While much is known in the case of Gaussian processes thanks to Talagrand’s celebrated majorizing measure theorem, 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, this gives the first steps towards the last missing piece in the study of suprema of general empirical processes.

The proof of Talagrand’s conjecture is combinatorial and easily leads to the proof of the Kahn-Kalai conjecture, an important question in probabilistic combinatorics and random graph theory.