Theses  Pure Mathematics and Mathematical Statistics
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Tripos models of Internal Set Theory
(20220129)This thesis provides a framework to make sense of models of E. Nelson’s Internal Set Theory (and hence of nonstandard analysis) in elementary toposes by exploiting the technology of tripos theory and Lawvere’s hyperdoctrines. ... 
Thick points of random walk and multiplicative chaos
(20211118)This thesis concentrates on the classical problem of understanding the multifractal properties of Brownian motion and random walk. Specifically, we will be interested in the set of thick points, that is points where the ... 
Kac's process and some probabilistic aspects of the Boltzmann equation
We consider a family of stochastic interacting particle systems introduced by Kac as a model for a spatially homogeneous gas undergoing elastic collisions, corresponding to the spatially homogeneous Boltzmann equation. We ... 
Asymptotic theory for Bayesian nonparametric inference in statistical models arising from partial differential equations
Partial differential equations (PDEs) are primary mathematical tools to model the behaviour of complex realworld systems. PDEs generally include a collection of parameters in their formulation, which are often unknown in ...