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UID:6eef014d6c63bc9b9a72d205b5c60787
CATEGORIES:Experimental Mathematics Seminar
CREATED:20230118T090917
SUMMARY:Thresholds
LOCATION:Zoom
DESCRIPTION:<p style="color: #000000; font-family: 'Times New Roman'; font-size: medium
 ; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; 
 text-indent: 0px; text-transform: none; white-space: normal; widows: 2; wor
 d-spacing: 0px;"><em>Abstract</em>: For a finite set X, a family F of subse
 ts of X is said to be increasing if any set A that contains B in F is also 
 in F. The p-biased product measure of F increases as p increases from 0 to 
 1, and often exhibits a drastic change around a specific value, which is ca
 lled a "threshold." Thresholds of increasing families have been of great hi
 storical interest and a central focus of the study of random discrete struc
 tures (e.g. random graphs and hypergraphs), with estimation of thresholds f
 or specific properties the subject of some of the most challenging work in 
 the area. In 2006, Jeff Kahn and Gil Kalai conjectured that a natural (and 
 often easy to calculate) lower bound q(F) (which we refer to as the “expect
 ation-threshold”) for the threshold is in fact never far from its actual va
 lue. A positive answer to this conjecture enables one to narrow down the lo
 cation of thresholds for any increasing properties in a tiny window. In par
 ticular, this easily implies several previously very difficult results in p
 robabilistic combinatorics such as thresholds for perfect hypergraph matchi
 ngs (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). In 
 this talk, I will present recent progress on this topic. Based on joint wor
 k with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.</p
 >
CONTACT:Jinyoung Park, Courant Institute, New York University.
X-EXTRAINFO:Zoom Link: https://rutgers.zoom.us/j/94346444480\npassword: The 20th Catala
 n number, alias (40)!/(20!*21!), alias 6564120420 ]
DTSTAMP:20260828T102325
DTSTART;TZID=America/New_York:20230202T170000
DTEND;TZID=America/New_York:20230202T180000
SEQUENCE:0
TRANSP:OPAQUE
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