Speaker: Jinyoung Park, Stanford University
Location: Warren Weaver Hall 512
Date: October 7, 2022, 1 p.m.
Host: Paul Bourgade
For a finite set X, a family F of subsets 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 called a "threshold." Thresholds of increasing families have been of great historical interest and a central 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 Kahn and Gil Kalai conjectured that a natural (and often easy to calculate) lower 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 any increasing properties in a tiny window. In particular, this easily implies several previously very difficult results in probabilistic combinatorics such as thresholds for perfect hypergraph matchings (Johansson–Kahn–Vu) and bounded-degree spanning trees (Montgomery). In this talk, I will present recent progress on this topic. Based on joint work with Keith Frankston, Jeff Kahn, Bhargav Narayanan, and Huy Tuan Pham.
Jinyoung Park is a Szegö Assistant Professor at Stanford University working in combinatorics. Previously, Park was a Member of the Institute for Advanced Study from 2020 to 2021. She received her Ph.D. in 2020 under the supervision of Jeff Kahn. Her doctoral work earned the 2022 Dissertation Prize from the Association for Women in Mathematics, and she will be awarded the 2023 Maryam Mirzakhani New Frontiers Prize "for contributions to the resolution of several major conjectures on thresholds and selector processes." Before she started her graduate studies, she worked as a mathematics teacher in secondary schools in Seoul from 2005 to 2011.
In-person attendance only available to those with active NYU ID cards.