Kevin Ford (University of Illinois at Urbana-Champaign) will present "Prime chains and applications." Abstract: A sequence of primes $p_1,...,p_k$ is a "prime chain" if $p_j|(p_{j+1}-1)$ for each $j$. For example: 3, 7, 29, 59, 709. We describe new estimates for counts of prime chains satisfying various properties, e.g. the number of chains with $p_k < x$ ($k$ variable) and the number of chains with $p_1=p$ and $p_k \le x$. We discuss some applications of these estimates, in particular the settling of a 50-year old conjecture of Erdos that $\phi(a)=\sigma(b)$ has infinitely many solutions ($\phi$ is Euler's function, $\sigma$ is the sum of divisors function). We also focus on the distribution of $H(p)$, the length of the longest chain ending at a given prime $p$. $H(p)$ is also the height of the "Pratt tree" for $p$, the tree structure of all chains ending at $p$. We give new, nontrivial bounds for $H(p)$, valid for almost all $p$, and settle a conjecture of Erdos, Granville, Pomerance and Spiro from 1990. We introduce and analyze a random model of the Pratt tree, based on branching random walks, which leads to some surprising conjectures about the distribution of $H(p)$. Finally, we give an application to groups with "perfect order subsets" and discuss various open problems in the area.