Pt. 1 – Asymptotic enumeration via graph containers and entropy | Jinyoung Park, NYU | IAS/PCMI скачать в хорошем качестве

Pt. 1 – Asymptotic enumeration via graph containers and entropy | Jinyoung Park, NYU | IAS/PCMI

Title: Asymptotic enumeration via graph containers and entropy - part 1 Presented to PCMI by Jinyoung Park, NYU Abstract: The container methods are powerful tools to bound the number of independent sets of graphs and hypergraphs, and they have been extremely influential in the area of extremal and probabilistic combinatorics. We will focus on more specialized graph containers due to Sapozhenko (1987), that specifically deal with independent sets in expanders. Entropy, first introduced by Shannon (1948), from information theory is a measure of the expected amount of information contained in a random variable. Entropy has seen lots of fascinating applications in a wide range of enumeration problems. In this series of lectures, we will discuss old and new applications of graph containers, entropy methods, and their combinations for various enumeration problems. Introductory-level knowledge of combinatorics and probability will be assumed. -- 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.