ICM 2026 Invited Section Lecture - Julian Sahasrabudhe

Value of the Information & Strength of the Argument

The lecture provides a clear overview of cutting-edge research, linking disparate problems through the lens of rare-event analysis. The argumentation is methodical: starting with classical results, highlighting limitations, and then introducing new techniques that overcome these limitations. The speaker effectively demonstrates the power of probability-based approaches even at extremely small scales, and his explanation of discrepancy theory as a tool for constructing polynomials is compelling. The reasoning is logical and well-structured, with technical details presented at an appropriate level for a specialized audience.

Scientific Rigor, Source Quality, Title Accuracy

The talk is scientifically rigorous, with careful definitions and references to known results. The speaker cites key authors and theorems (Erdős, Lovász, Spencer, Beck, Rudin-Shapiro) and explains the context of each result. No external links are provided in the description, but the talk itself serves as a reliable survey from an expert. The title matches the content closely. The lecture is an invited section lecture at the ICM, indicating high peer recognition.

Title accurately reflects the content: an invited lecture at ICM 2026 on probabilistic combinatorics, focusing on exponentially small probability scales.

8/10

The lecture is a survey by a leading researcher in probabilistic combinatorics, covering recent developments including his own work. The presentation is rigorous, with clear definitions and references to established results. No promotional or speculative content.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:30Introduction to the probabilistic method and the concept of exponentially small scales.
• 02:00Overview of Ramsey numbers and the use of random colorings to prove lower bounds.
• 05:00Discussion of techniques for tackling exponentially unlikely events: Lovász local lemma, Beck's method, random processes, large deviation theory.
• 08:00Introduction of flat Littlewood polynomials, the central problem of the talk, and the failure of random polynomials.
• 12:00Connection to discrepancy theory and Spencer's theorem, leading to a rich class of polynomials satisfying the upper bound.
• 18:00Lovász's random process proof via sticky Brownian motion for finding discrepancy solutions.
• 22:00Discussion of lower bound approaches for flat Littlewood polynomials, including tensorization and partition strategies.
• 30:00Mention of related recent results: triangle-free process, sphere packing in large dimensions, and singularity of random matrices.

• Erdős's lower bound for Ramsey numbers (via probabilistic method) — Discussed as a classic application of the probabilistic method to prove that R(k) > 2^(k/2).
• Lovász Local Lemma — Mentioned as a key technique for handling rare events in probability spaces.
• Beck's method (partial coloring) and Spencer's theorem — Used to show existence of discrepancy O(√n) for matrices, and applied to construct many polynomials with bounded sup norm.
• Rudin-Shapiro polynomials — Classical examples of polynomials with small sup norm, used as a starting point for the one-sided bound.