BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Lecture 1 | The Hypergraph Container Method - Jozsef Balogh (Unive rsity of Illinois at Urbana-Champaign) DTSTART:20240731T133000Z DTEND:20240731T143000Z UID:TALK219241@talks.cam.ac.uk DESCRIPTION:In this survey we describe a recently-developed technique for bounding the number (and controlling the typical structure) of finite obje cts with forbidden substructures. This technique exploits a subtle cluster ing phenomenon exhibited by the independent sets of uniform hypergraphs wh ose edges are sufficiently evenly distributed\; more precisely\, it provid es a relatively small family of &lsquo\;containers&rsquo\; for the indepen dent sets\, each of which contains few edges. We attempt to convey to the reader a general high-level overview of the method\, focusing on a small n umber of illustrative applications in areas such as extremal graph theory\ , Ramsey theory\, additive combinatorics\, and discrete geometry\, and avo iding technical details as much as possible.In Lecture 1\, besides motivat ions\, we will focus on the graph container method\, in Lecture 2 several applications of the Hypergraph Container Lemma will be discussed\, and in Lecture 3\, the sketch of the proof of the \; Hypergraph Container Lem ma will be discussed. Certainly\, some adjustment based on Lecture 1 will be done on the later lectures. LOCATION:External END:VEVENT END:VCALENDAR