2014 CMS Summer Meeting
University of Manitoba, June 6 - 9, 2014
In percolation much work has been done on the critical probability, since it is strictly between $0$ and $1$ in every non-trivial model; however, in $r$-neighbour bootstrap percolation the critical probability is of no interest, since every lattice has critical probability $0$ or $1$.
Recently, Smith, Uzzell and the lecturer introduced a wide-ranging extension of $r$-neighbour bootstrap percolation: they initiated the study of completely general monotone, local, and homogeneous cellular automata in a random environment. Among other results, they classified these $\mathcal U$-percolation models into three types, and proved results about the phase transition in two of them. The phase transition in the third type has been clarified by Balister, Przykucki, Smith and the lecturer: in particular, they have shown that in ${\mathbb Z}^2$ these processes have non-trivial critical probabilities. These results have reopened the study of critical probabilities in `generalized bootstrap processes' on ${\mathbb Z}^2$.
In the lecture I shall survey some classical results in percolation and bootstrap
percolation, and will sketch some of the recent results on monotone cellular automata
obtained by Balister, Duminil-Copin, Gunderson, Holmgren, Morris, Przykucki, Smith, Uzzell, and myself.
In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates.
The plan for the talk is (1) a brief history of analytic capacity and its applications, (2) a practical method for rigorous computation of analytic capacity, and (3) the hunt for a counterexample to the subadditivity problem. Parts (2) and (3) are based on joint work with Malik Younsi.