arXiv subjects

Adam Shaw

Publications and source records attributed to Adam Shaw.

6 recordsLinked to original sources

arXiv2026

Topological Simplification in Predictive Coding Networks

We study the topology of learned representations in predictive coding networks (PCNs), a neuro-inspired bidirectional architecture, using a quantitative layer-wise persistent homology analysis. We train well-performing PCNs on a synthetic classification dataset ($\geq 99.9\%$ test accuracy) and on MNIST ($\geq 95\%$ test accuracy), and measure how topological features change across layers for different architectures and activation functions. We find that smaller PCNs collapse connected components across layers earlier than larger models (Spearman $\unicode{x1D70C} \in [0.72, 0.79]$ across activations), with model size measured as the sum of hidden-layer widths. We also observe a strong negative correlation ($\unicode{x1D70C} = -0.58$) between the depth at which simplification occurs and reconstruction error; i.e., architectures that simplify later reconstruct better. Finally, a seed-level bootstrap comparison across architectures and activations shows that PCNs consistently collapse connected components later than matched MLPs, with an average difference of $3.6$ layers. These results suggest that persistent homology offers a useful quantitative lens on the compression--reconstruction tradeoff in PCNs, and that both model capacity and the recurrent, bidirectional dynamics of predictive coding inference shape when this tradeoff is resolved across layers.

cs.LG↗

arXiv2026

Kahler decoupling for Kerr perturbations

The Euclidean Kerr metric is conformal, in two distinct ways, to a Kahler metric, with conformal factors determined by the repeated eigenvalue of the two chiral halves of the Weyl curvature. A Lorentzian analogue holds, where the conformally related metric is complex but retains key features of Kahler geometry. We show that this hidden Kahler structure provides a geometric explanation for the existence of decoupled equations for curvature scalars, such as the Teukolsky equations. The essential mechanism is that, on a Kahler background, self-dual 2-forms are parallel with respect to a natural covariant derivative, so differential operators acting on them preserve their decomposition and do not mix components. In this way, decoupling is seen to be a direct consequence of Kahler geometry. We make this mechanism explicit in two ways. First, we show that the spin-k Teukolsky operator can be obtained from a Laplace-type operator associated with the Kahler metric by a similarity transformation. Second, for electromagnetic perturbations, we use the conformal invariance of Maxwell's equations delta F = 0 to show that they imply d delta F = 0, where delta is the co-differential of the Kahler metric. This operator automatically decouples, and the resulting equations for the extremal components coincide with the spin-one Teukolsky equations.

gr-qc↗

arXiv2026

General Relativity via differential forms -- explorations in Plebanski's Formalism for GR

This thesis studies general relativity (GR) using chiral formulations, which take advantage of the decomposition of the four-dimensional Lorentz group into self-dual and anti-self-dual sectors. Within this framework, GR can be expressed using Plebanski's formulation, where the basic variables are triples of 2-forms rather than a metric, or alternatively through pure connection approaches. These viewpoints expose additional structure in Einstein's equations (EEs) and offer new analytical and numerical tools. Part I develops the geometric foundations using fibre bundles, where the 2-forms arise as soldering forms on an SO(3,C) bundle. Part II investigates the linearised form of EEs in the chiral setting, with particular attention to their gauge fixings. Part III extends this analysis to the nonlinear regime, and also examines the complex-geometric structure underlying black hole spacetimes. The final part turns to numerical relativity, exploring evolution schemes built from the chiral formulations and their associated gauge choices.

gr-qc↗

arXiv2025

Plebanski complex

As is very well-known, linearisation of the instanton equations on a 4-manifold gives rise to an elliptic complex of differential operators, the truncated (twisted) Hodge complex $\Lambda^0(\mathfrak{g}) \to \Lambda^1(\mathfrak{g})\to \Lambda^2_+(\mathfrak{g})$. Moreover, the linearisation of the full YM equations also fits into this framework, as it is given by the second map followed by its adjoint. We define and study properties of what we call the Pleba\'nski complex. This is a differential complex that arises by linearisation of the equations implying that a Riemannian 4-manifold is hyper-K\"ahler. We recall that these are most naturally stated as the condition that there exists a perfect $\Sigma^i\wedge \Sigma^j\sim\delta^{ij}$ triple $\Sigma^i, i=1,2,3$ of 2-forms that are closed $d\Sigma^i=0$. The Riemannian metric is encoded by the 2-forms $\Sigma^i$. We show that what results is an elliptic differential complex $TM \to S\to E\times \Lambda^1 \to E$, where $S$ is the tangent space to the space of perfect triples, and $E=\mathbb{R}^3$. We also show that, as in the case with instanton equations, the full Einstein equations $Ric=0$ also fit into this framework, their linearisation being given by the second map followed by its adjoint. Our second result concerns the elliptic operator that the Pleba\'nski complex defines. In the case of the instanton complex, operators appearing in the complex supplemented with their adjoints assemble to give the Dirac operator. We show how the same holds true for the Pleba\'nski complex. Supplemented by suitable adjoints, operators assemble into an elliptic operator that squares to the Laplacian and is given by the direct sum of two Dirac operators.

math.DG↗

arXiv2024

Kerr metric from two commuting complex structures

The main aim of this paper is to simplify and popularise the construction from the 2013 paper by Apostolov, Calderbank, and Gauduchon, which (among other things) derives the Plebanski-Demianski family of solutions of GR using ideas of complex geometry. The starting point of this construction is the observation that the Euclidean versions of these metrics should have two different commuting complex structures, as well as two commuting Killing vector fields. After some linear algebra, this leads to an ansatz for the metrics, which is half-way to their complete determination. Kerr metric is a special 2-parameter subfamily in this class, which makes these considerations directly relevant to Kerr as well. This results in a derivation of the Kerr metric that is self-contained and elementary, in the sense of being mostly an exercise in linear algebra.

gr-qc↗

arXiv2022

Weyl Curvature Evolution System for GR

Starting from the chiral first-order pure connection formulation of General Relativity, we put the field equations of GR in a strikingly simple evolution system form. The two dynamical fields are a complex symmetric tracefree 3x3 matrix Psi, which encodes the self-dual part of the Weyl curvature tensor, as well as a spatial SO(3,C) connection A. The right-hand sides of the evolution equations also contain the triad for the spatial metric, and this is constructed non-linearly from the field Psi and the curvature of the spatial connection A. The evolution equations for this pair are first order in both time and spatial derivatives, and so simple that they could have been guessed without a computation. They are also the most natural generalisations of the equations one obtains in the case of the chiral description of Maxwell's theory. We also determine the modifications of the evolution system needed to enforce the "constraint sweeping", so that any possible numerical violation of the constraints present becomes propagating and gets removed from the computational grid.

gr-qc↗