C2

We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.
 

A non-parametric test for dependence between sets of random variables based on the entropy rate is proposed. The test has correct size, unit asymptotic power, and can be applied to test setwise cross sectional and serial dependence. Using Monte Carlo experiments, we show that the test has favourable small-sample properties when compared to other tests for dependence. The ‘trick’ of the test relies on using universal codes to estimate the entropy rate of the stochastic process generating the data, and simulating the null distribution of the estimator through subsampling. This approach avoids having to estimate joint densities and therefore allows for large classes of dependence relationships to be tested. Potential economic applications include model specification, variable and lag selection, data mining, goodness-of-fit testing and measuring predictability.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.

We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles.  We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.