Pt.5–From Sunflowers to Thresholds | Shachar Lovett, University of California, San Diego | IAS/PCMI скачать в хорошем качестве

Pt.5–From Sunflowers to Thresholds | Shachar Lovett, University of California, San Diego | IAS/PCMI

From Sunflowers to Thresholds - part 5 Presented to PCMI by Shachar Lovett, University of California, San Diego Abstract: A sunflower is a basic notion in combinatorics. A family of sets is called a sunflower if their pairwise intersections are all the same. The sunflower conjecture of Erdos and Rado is a fundamental open problem in combinatorics, which asks for the minimal size of a family of sets that must contain a sunflower of a given size. A few years ago, major progress was made on the conjecture, leveraging surprising connections to computational complexity. In follow up works, even more surprising connections were found to threshold phenomena, which eventually led to the resolution of the Kahn-Kalai conjecture, a major open problem in the area. In this series of talks, I will describe these results, with an emphasis on how connections between different fields of mathematics and theoretical computer science played a pivotal role. No specific prerequisites are needed except for mathematical maturity. -- Lecture notes & problem sets https://www.ias.edu/pcmi/pcmi-2025-gs... PCMI 2025 GSS Lecture Notes and Problem Sets - IAS/Park City Mathematics Institute -- PCMI 2025 Research Topic: Probabilistic and Extremal Combinatorics Organized by Julia Böttcher (LSE), Jacob Fox (Stanford University), Penny Haxell (University of Waterloo), Robert Morris (IMPA), and Wojciech Samotij (Tel Aviv University). Extremal and probabilistic combinatorics are two central branches of contemporary discrete mathematics. The first of these two branches studies how large (or how small) a discrete structure can be given that it satisfies a certain set of restrictions; the second investigates random combinatorial objects using a blend of combinatorial methods and tools of probability theory. These two fields have been growing at a stunning rate over the last few decades and are nowadays considered to be an important part of mainstream mathematical research. – The aim of the planned summer graduate program at PCMI is to provide in-depth introduction to several preeminent themes and methods in extremal and probabilistic combinatorics, with particular emphasis on strong connections of these fields with other areas of mathematics such as analysis, geometry, number theory, statistical physics, and theoretical computer science. The core of the program will be nine graduate mini-courses taught by a diverse group of leading researchers in the field renowned for their clear and engaging lecturing styles. In parallel, we plan thematic workshops aimed at more senior researchers as well as activities for undergraduate students.