## Extremal Combinatorics

**Date:**
21 January - 16 May 2025

**Location: ** Simons Laufer Mathematical Research Institute

**Event type:**
Extended Format

**Organisers: ** David Conlon (CalTech), Jacob Fox (Stanford), Penny Haxell (Waterloo), Janos Pach (Rényi Institute), Maya Stein (Chile), Andrew Suk (UCSD)

**Website:**www.slmath.org/programs/375

Extremal combinatorics concerns itself with problems about how large or small a finite collection of objects can be while satisfying certain conditions. Questions of this type arise naturally across mathematics, so this area has close connections and interactions with a broad array of other fields, including number theory, group theory, model theory, probability, statistical physics, optimization, and theoretical computer science.

The area has seen huge growth in the twenty-first century and, particularly in recent years, there has been a steady stream of solutions to important longstanding problems and many powerful new methods have been introduced. These advances include improvements in absorption techniques which have facilitated the proof of the existence of designs and related objects, the breakthrough on the sunflower conjecture whose further development eventually led to the proof of the Kahn–Kalai conjecture in discrete probability and the discovery of interactions between spectral graph theory and the study of equiangular lines in discrete geometry. These and other groundbreaking advances will be the central theme of this semester program.

Professor Gabor Tardos (Alfréd Rényi Institute) has been appointed as a Clay Senior Scholar to participate in this program.