arXiv subjects
Publications and source records attributed to Linden Disney-Hogg.
We demonstrate a method to prove that a theta characteristic on Riemann surface $S$ invariant under the action of $G \leq \mathrm{Aut}(S)$ has even parity by considering the representation theory of the double cover $\tilde{G} = 2 \cdot G$. We apply this to certain Hurwitz curves, and prove a recent conjecture of Broughton \& Disney-Hogg.
Given a finite group $G$ acting orientably on a surface $S$ of genus $σ\geq 2$, the group $G$ acts faithfully on the homology group $H_{1}(S;\mathbb{Z})$, preserving the symplectic intersection form. The action on $S$ and the homology is determined by a \emph{generating vector}, a tuple of elements of $G$, generating $G$ and satisfying certain properties. In this note we show how to compute the homology representation, using the generating vector, when $S/G$ has genus 0 and the genus is suitably low. A $2σ\times 2σ$ representing matrix can be determined for any element in the group, usually for a small set of generators. The matrices are computed with respect to an auto-generated basis for the cellular homology of $S$, using a regular $CW$ structure on $S$, derived from the $G$ action. We demonstrate the application of these results by computing \emph{invariant theta characteristics} of the Riemann surfaces $S$ with the algorithm implemented using Sage.
We develop and prove new geometric and algebraic characterisations for locations of constituent skyrmions, as well as their signed multiplicity, using Sutcliffe's JNR ansatz. Some low charge examples, and their similarity to BPS monopoles, are discussed. In addition, we provide Julia code for the further numerical study and visualisation of JNR skyrmions.
We show that any $SU(2)$ BPS monopole of charge $k$ with rotational spatial dihedral symmetry is gauge equivalent to the Nahm data obtained from affine Toda equations of $C_l^{(1)}$ type when $k=2l$ or $A_{2(l-1)}^{(2)}$ type when $k=2l-1$.
We present an algorithm which uses Fujiwara's inequality to bound algebraic functions over ellipses of a certain type, allowing us to concretely implement a rigorous Gauss-Legendre integration method for algebraic functions over a line segment. We consider path splitting strategies to improve convergence of the method and show that these yield significant practical and asymptotic benefits. We implemented these methods to compute period matrices of algebraic Riemann surfaces and these are available in SageMath.
The theta characteristics on a Riemann surface are permuted by the induced action of the automorphism group, with the orbit structure being important for the geometry of the curve and associated manifolds. We describe two new methods for advancing the understanding of these orbits, generalising existing results of Kallel & Sjerve, allowing us to establish the existence of infinitely many curves possessing a unique invariant characteristic as well as determine the number of invariant characteristics for all Hurwitz curves with simple automorphism group. In addition, we compute orbit decompositions for a substantial number of curves with genus $\leq 9$, allowing the identification of where current theoretical understanding falls short and the potential applications of machine learning techniques.
We classify all possible charge-3 monopole spectral curves with non-trivial automorphism group and within these identify those with elliptic quotients. By focussing on elliptic quotients the transcendental constraints for a monopole spectral curve become ones regarding periods of elliptic functions. We construct the Nahm data and new monopole spectral curves with $D_6$ and $V_4$ symmetry, the latter based on an integrable complexification of Euler's equations, and for which energy density isosurfaces are plotted. Extensions of our approach to higher charge and hyperbolic monopoles are discussed.
Bring's curve, the unique Riemann surface of genus-4 with automorphism group $S_5$, has many exceptional properties. We review, give new proofs of, and extend a number of these including giving the complete realisation of the automorphism group for a plane curve model, identifying a new elliptic quotient of the curve and the modular curve $X_0(50)$, providing a complete description of the orbit decomposition of the theta characteristics, and identifying the unique invariant characteristic with the divisor of the Szëgo kernel. In achieving this we have used modern computational tools in Sagemath, Macaulay2, and Maple, for which notebooks demonstrating calculations are provided.
This is an English translation of the paper "Ueber die algebraischen Curven von den Geschlechtern $p = 4$, $5$ und $6$, welche eindeutigen Transformationen in sich besitzen" by Anders Wiman, Bihang till Kongl. Svenska vetenskaps-akademiens handlingar 21 (1895): 1-41. The article, originally written in German, classifies non-hyperelliptic curves which have non-trivial automorphism groups via classical methods in algebraic geometry.