I am a fourth year PhD student under the supervision of Prof. John Greenlees. I am studying Homotopy Theory, which is a subfield of Algebraic Topology. At the moment I am thinking about localizations and completions of nilpotent spaces, as well as learning about the subject more broadly.
Model structures on topological spaces (First year PhD project) - overview of model categories, ordinals and the small object argument, introduction to simplicial sets, geometric realization takes Kan fibrations to Hurewicz fibrations, derivation of the Quillen model structure on simplicial sets and the q, h and m-model structures on spaces.
Finitely generated nilpotent spaces - a nilpotent space is equivalent to a CW complex with finite skeleta iff its homotopy groups are finitely generated iff its homology groups are finitely generated.
Simplicial Spaces and Fibrations - geometric realization takes 'locally trivial' maps of simplicial spaces to Hurewicz fibrations. We deduce that the realization of the orbit map EG to BG is an Hurewicz fibration.
Papers and Preprints
Completion preserves homotopy fibre squares of connected nilpotent spaces - ArXiv preprint, 2022
A double coset formula for the genus of a nilpotent group - ArXiv preprint, 2022