Braden, Harry

Figueroa-O'Farrill, José

Disney-Hogg, Alec Linden

Engineering and Physical Sciences Research Council (EPSRC)

University of Edinburgh

2024-02-12T14:01:07Z

2024-02-12T14:01:07Z

2024-02-12

This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions.

https://hdl.handle.net/1842/41435

https://github.com/DisneyHogg/Riemann_Surfaces_and_Monopoles

http://dx.doi.org/10.7488/era/4167

en

The University of Edinburgh

H. W. Braden and L. Disney-Hogg, Bring’s curve: Old and new, arXiv:2208.13692, 2022.

N. Bruin, L. Disney-Hogg, and W. E. Gao, Rigorous numerical integration of algebraic functions, arXiv:2208.12377, 2022.

L. Disney-Hogg, Abel-Jacobi in SageMath, https://github.com/ DisneyHogg/Abel-Jacobi-in-SageMath, 2021.

L. Disney-Hogg, A. Beckett, and I. Deutsch, An English translation of A. Wiman’s “on the algebraic curves of genus p = 4, 5 and 6, which posses unambiguous transformations into themselves”, arXiv.2204.01656, 2022.

Towards a classification of charge-3 monopoles with symmetry Braden, H. W. & Disney-Hogg, L., 31 Aug 2023, In: Letters in mathematical physics. 113, 4, 87.

Riemann surfaces

Symmetries of Riemann Surfaces

Magnetic Monopoles

symmetry in elucidating structure

automorphisms of algebraic curves

Bring’s curve

constructing magnetic monopole solutions

Hitchin conditions

Symmetries of Riemann surfaces and magnetic monopoles

Thesis or Dissertation

Doctoral

PhD Doctor of Philosophy