Jeff Kahn, Rutgers University

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(The colloquium password will be distributed to Princeton University and IAS members. We ask that you do not share this password. If you would like to be included in the colloquium and are not a member of either institution, please email the organizer Casey Kelleher ( with an email requesting to participate which introduces yourself, your current affiliation and stage in your career.)

Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some [here finite] set X, and the "threshold question" for such an F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F?  For example:  about how many random edges from the complete graph on n vertices are typically required to produce a Hamiltonian cycle?) We will try to give some sense of this area and then focus on a few recent highlights.