The ACO Seminar (2022–2023)

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 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 — Talagrand’s unfulfilled dreams.

The proof of Talagrand’s conjecture is combinatorial; one of its key ideas leads to the proof of the Kahn-Kalai conjecture on thresholds, a fundamental conjecture in probabilistic combinatorics and random graph theory. Time permitting, I will also discuss related problems, such as sunflowers in set systems, several problems and recent results on computation of thresholds, including joint work with Vishesh Jain resolving a conjecture of Luria and Simkin on threshold for containment of Latin squares and Steiner triple systems, and other related conjectures of Talagrand.

Based on joint works with Jinyoung Park, Vishesh Jain, Ashwin Sah, Mehtaab Sawhney and Michael Simkin.