Pruning processes $(mathcal{F}(theta),thetageq 0)$ have been studied separately for Galton-Watson trees and for Levy trees/forests. We establish here a limit theory that strongly connects the two studies. This solves an open problem by Abraham and De
lmas, also formulated as a conjecture by Lohr, Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson forests $mathcal{F}_n$, $ngeq 1$, in the domain of attraction of a Levy forest $mathcal{F}$, suitably scaled pruning processes $(mathcal{F}_n(theta),thetageq 0)$ converge in the Skorohod topology on cadlag functions with values in the space of (isometry classes of) locally compact real trees to limiting pruning processes. We separately treat pruning at branch points and pruning at edges. We apply our results to study ascension times and Kesten trees and forests.
The full moduli space M of a class of N=1 supersymmetric gauge theories is studied. For gauge theories living on a stack of D3-branes at Calabi-Yau singularities X, M is a combination of the mesonic and baryonic branches, the former being the symmetr
ic product of X. In consonance with the mathematical literature, the single brane moduli space is called the master space F. Illustrating with a host of explicit examples, we exhibit many algebro-geometric properties of the master space such as when F is toric Calabi-Yau, behaviour of its Hilbert series, its irreducible components and its symmetries. In conjunction with the plethystic programme, we investigate the counting of BPS gauge invariants, baryonic and mesonic, using the geometry of F and show how its refined Hilbert series not only engenders the generating functions for the counting but also beautifully encode ``hidden global symmetries of the gauge theory which manifest themselves as symmetries of the complete moduli space M for arbitrary number of branes.
