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We consider maps on a surface of genus g with all vertices of degree at least three and positive real lengths assigned to the edges. In particular, we study the family of such metric maps with fixed genus g and fixed number n of faces with circumferences α 1 , ... , α n and a β -irreducibility constraint, which roughly requires that all contractible cycles have length at least β . Using recent results on the enumeration of discrete maps with an irreducibility constraint, we compute the volume V g , n (β) (α 1 , ... , α n) of this family of maps that arises naturally from the Lebesgue measure on the edge lengths. It is shown to be a homogeneous polynomial in β , α 1 , ... , α n of degree 6 g - 6 + 2 n and to satisfy string and dilaton equations. Surprisingly, for g = 0 , 1 and β = 2 π the volume V g , n (2 π) is identical, up to powers of two, to the Weil–Petersson volume V g , n WP of hyperbolic surfaces of genus g and n geodesic boundary components of length L i = α i 2 - 4 π 2 , i = 1 , ... , n . For genus g ≥ 2 the identity between the volumes fails, but we provide explicit generating functions for both types of volumes, demonstrating that they are closely related. Finally we discuss the possibility of bijective interpretations via hyperbolic polyhedra. [ABSTRACT FROM AUTHOR]

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall and Miermont in Ann Probab 39:1-69, 2011). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in Bertoin et al. (Ann Probab, accepted). [ABSTRACT FROM AUTHOR]

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2 + 1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly under a certain assumption on the kinematic Hilbert space, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then discuss the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity Shape Dynamics Hamiltonian in higher dimensions. [ABSTRACT FROM AUTHOR]

We consider two interpretations for what it means for test data to demonstrate correctness. For each interpretation, we examine under what conditions data sufficient to demonstrate correctness exists, and whether it can be automatically detected and/or generated. We establish the relation between these questions and the problem of deciding equivalence of two programs. [ABSTRACT FROM AUTHOR]