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UID:6eef014d6c63bc9b9a72d205b5c60787
CATEGORIES:Experimental Mathematics Seminar
CREATED:20230118T090917
SUMMARY:Thresholds
LOCATION:Zoom
DESCRIPTION:Abstract: For a finite set X, a family F of subsets of X is said to be incr
easing 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 dr
astic change around a specific value, which is called a "threshold." Thresh
olds of increasing families have been of great historical interest and a ce
ntral focus of the study of random discrete structures (e.g. random graphs
and hypergraphs), with estimation of thresholds for specific properties the
subject of some of the most challenging work in the area. In 2006, Jeff Ka
hn and Gil Kalai conjectured that a natural (and often easy to calculate) l
ower bound q(F) (which we refer to as the “expectation-threshold”) for the
threshold is in fact never far from its actual value. A positive answer to
this conjecture enables one to narrow down the location of thresholds for a
ny increasing properties in a tiny window. In particular, this easily impli
es several previously very difficult results in probabilistic combinatorics
such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) an
d bounded-degree spanning trees (Montgomery). In this talk, I will present
recent progress on this topic. Based on joint work with Keith Frankston, Je
ff Kahn, Bhargav Narayanan, and Huy Tuan Pham.\n
X-ALT-DESC;FMTTYPE=text/html:*Abstract*: 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.

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:20231004T034152
DTSTART;TZID=America/New_York:20230202T170000
DTEND;TZID=America/New_York:20230202T180000
SEQUENCE:0
TRANSP:OPAQUE
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