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The Universality Conjecture of Bollobás, Duminil-Copin, Morris and Smith states that every $d$-dimensional monotone cellular automaton is a member of one of $d+1$ universality classes, which are characterized by their behaviour on sparse random sets. More precisely, it states that if sites are initially infected independently with probability $p$, then the expected infection time of the origin is either infinite, or is a tower of height $r$ for some $r \in \{1,...,d\}$.

In this talk I will state a theorem which proves the conjecture, and moreover determines the value of $r$ for every model. I will also attempt to motivate this theorem by discussing some interesting (and well-studied) special cases, and some potential applications to non-monotone models such as the Ising model of ferromagnetism, and kinetically constrained models of the liquid-glass transition.

Joint work with Paul Balister, Béla Bollobás and Paul Smith.

There will be a reception before the seminar, at 3:30 pm in the Alder room.