We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension $i$ tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least $i-g+1$ genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Diaz, Faber, Getzler, Ionel, Looijenga, Pandharipande, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by the number of rational components.

We use random walks to show that the Basilica group is amenable and thus answering an open question of Grigorchuk and Żuk [9]. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.

We complete the explicit study of a three-fold divisorial contraction whose exceptional divisor contracts to a point by treating the case where the point downstairs is a singularity of index $n\ge 2$. We prove that if this singularity is of type c$A/n$, then any such contraction is a weighted blowup; and that if otherwise, then $f$ is either a weighted blowup of a singularity of type c$D/2$ embedded into a cyclic quotient of a smooth five-fold, or a contraction with discrepancy $1/n$, 1, or 2. Every such exceptional case of discrepancy 1 or 2 has an example. The erratum to our previous article [13] appears in the appendix.

We study some path transformations related to Pitman's theorem [28, Th. 1.3] on Brownian motion and the three-dimensional Bessel process. We relate these to the Littelmann path model (see [22]) and give applications to representation theory and to Brownian motion in a Weyl chamber

Let ${\displaystyle {\mathrm{Rat}}_d}$ denote the space of holomorphic self-maps of $P^1$ of degree $d\ge 2$, and let ${\mu }_f$ be the measure of maximal entropy for $f\in {\mathrm{Rat}}_d$. The map of measures $f\to {\mu }_f$ is known to be continuous on ${\mathrm{Rat}}_d$, and it is shown here to extend continuously to the boundary of ${\mathrm{Rat}}_d$ in ${\stackrel{̲}{\mathrm{Rat}}}_d\simeq P{H}^0(P^1\times P^1,O(d,1))\simeq P^{2d+1}$, except along a locus $I(d)$ of codimension $d+1$. The set $I(d)$ is also the indeterminacy locus of the iterate map $f\to f^n$ for every $n\ge 2$. The limiting measures are given explicitly, away from $I(d)$. The degenerations of rational maps are also described in terms of metrics of nonnegative curvature on the Riemann sphere; the limits are polyhedral