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Free groups and the axiom of choice
(University of CambridgeDepartment of Pure Mathematics and Mathematical Statistics, 2016-01-05)
The Nielsen–Schreier theorem states that subgroups of free groups are free. As all of its proofs use the Axiom of Choice, it is natural to ask whether the theorem is equivalent to the Axiom of Choice. Other questions arise ...
Deformations of Cayley submanifolds
(University of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsTrinity College, 2016-03-01)
Cayley submanifolds of R^8 were introduced by Harvey and Lawson as an instance of calibrated submanifolds, extending the volume-minimising properties of complex submanifolds in Kähler manifolds. More generally, Cayley ...
Spectral methods and computational trade-offs in high-dimensional statistical inference
(Department of Pure Mathematics and Mathematical Statistics, University of CambridgeUniversity of CambridgeDepartment of Pure Mathematics and Mathematical StatisticsFaculty of MathematicsSt John's College, 2016-10-04)
Spectral methods have become increasingly popular in designing fast algorithms for modern highdimensional datasets. This thesis looks at several problems in which spectral methods play a central role. In some cases, we ...
Standard model physics from an algebra?
(2015)
This thesis constitutes a first attempt to derive aspects of standard model particle physics from little more than an algebra. Here, we argue that physical concepts such as particles, causality, and irreversible time may ...
First-order gradient regularisation methods for image restoration: reconstruction of tomographic images with thin structures and denoising piecewise affine images
(University of CambridgeDepartment of Applied Mathematics and Theoretical PhysicsCentre for Mathematical SciencesJesus College, 2016-02-02)
The focus of this thesis is variational image restoration techniques that involve novel non-smooth first-order gradient regularisers: Total Variation (TV) regularisation in image and data space for reconstruction of thin ...
The power spectrum and bispectrum of inflation and cosmic defects
(University of CambridgeDepartment of Applied Mathematics and Theoretical PhysicsGonville and Caius College, 2016-06-28)
Much of the recent progress in cosmology has come from studying the power spectrum of the cosmic microwave background (CMB). The latest results from the Planck satellite confirmed that the inflationary paradigm with the ...
Generalised nonlinear stability of stratified shear flows: adjoint-based optimisation, Koopman modes, and reduced models
(Department of Applied Mathematics and Theoretical Physics, University of CambridgeUniversity of CambridgeDepartment of Applied Mathematics and Theoretical PhysicsSt John's College, 2016-10-04)
In this thesis I investigate a number of problems in the nonlinear stability of density stratified plane Couette flow. I begin by describing the history of transient growth phenomena, and in particular the recent application ...