This year’s winter school is taking place on Jan 6-10, 2025. All information, including lecture notes, recordings will be available on the course homepage: https://www.ibs.re.kr/ecopro/winter-2025/
Lecturer: Huy Pham (IAS)
Recent developments in probabilistic combinatorics: New synergies
Recent developments in probabilistic combinatorics have brought about synergistic interactions across multiple disciplines, including extremal combinatorics, probability theory and theoretical computer science. This mini-course aims to give an introduction to some of these developments and interactions.
In the first part of the mini-course, we will discuss the Kahn-Kalai conjecture on thresholds and expectation thresholds and its close relation to Talagrand’s selector process conjecture in probability theory. We will also touch on several subsequent developments of the proof technique in extremal combinatorics, probability theory and theoretical computer science.
The Kahn-Kalai conjecture opens up a new avenue on thresholds of interesting properties in random graph models. Yet, while it relates thresholds to the expectation thresholds, estimation of the latter remains a highly challenging task in general. Recent developments have found close connections between this task and previous fundamental developments in extremal combinatorics, such as the regularity method. In the second part of the mini-course, I will briefly overview classical aspects of the regularity method before discussing its new applications to thresholds and other problems in probabilistic combinatorics.
