The Homogeneous Geometries of Real Hyperbolic Space

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components.     

Noetherianity of some degree two twisted commutative algebras

The resolutions of determinantal ideals exhibit a remarkable stability property: for fixed rank but growing dimension, the terms of the resolution stabilize (in an appropriate sense). One may wonder if other sequences of ideals or modules over coordinate rings of matrices exhibit similar behavior. We show that this is indeed the case. In fact, our main theorem is more fundamental in nature: It states that certain large algebraic structures (which are examples of twisted commutative algebras) are noetherian. These are important new examples of large noetherian algebraic structures, and ones that are in some ways quite different from previous examples.     

On using compressibility to detect when slime mould completed computation

Slime mould Physarum polycephalum is a single cell visible by an unaided eye. The slime mould optimizes its network of protoplasmic tubes in gradients of attractants and repellents. This behavior is interpreted as computation. Several prototypes of the slime mould computers were designed to solve problems of computation geometry, graphs, transport networks, and to implement universal computing circuits. Being a living substrate, the slime mould does not halt its behavior when a task is solved but often continues foraging the space thus masking the solution found. We propose to use temporal changes in compressibility of the slime mould patterns as indicators of the halting of the computation. Compressibility of a pattern characterizes the pattern's morphological diversity, that is, a number of different local configurations. At the beginning of computation the slime explores the space, thus generating less compressible patterns. After gradients of attractants and repellents are detected the slime spans data sites with its protoplasmic network and retracts scouting branches, thus generating more compressible patterns. We analyze the feasibility of the approach on results of laboratory experiments and computer modelling. © 2015 Wiley Periodicals, Inc. Complexity 21: 162–175, 2016     

The school class allocation problem—maximising friendship requests subject to diversity requirements

Friendship groups are an important element in both the social and academic well-being of school students, particularly in their younger years. Each year, school management has to assign students to a particular class in their new grade in junior school, and in doing so make and break friendship preferences as other criteria need to be satisfied in order to ensure diversity. This ongoing administrative task is time consuming and does not always result in the most equitable allocation. Coupled with the lack of transparency of the process, this can lead to teacher and parent frustration. This paper formulates and solves a mathematical programming model for this problem and shows that better solutions can likely be found in a fraction of the time using software that is freely available.     

The Szegö kernel on a class of non-compact CR manifolds of high codimension

We generalize Nagel’s formula for the Szegö kernel and use it to compute the Szegö kernel on a class of non-compact CR manifolds whose tangent space decomposes into one complex direction and several totally real directions. We also discuss the control metric on these manifolds and relate it to the size of the Szegö kernel.