School of Mathematics and Statistics - Theses
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ItemGiving Daggers to Higher Cats: Generalised Quasi Operads, Astroidal Sets, and a Surface Operad( 2020)This thesis contains my work on various generalisations of infinity operads, as well as an example inspired by Topological Quantum Field Theories (TQFTs). The main result is a proof of the equivalence between the Segal and strict inner Kan conditions for graphical sets. There are two different generalisations of infinity operads explored herein. Firstly, there are dagger categories. Inspired by Hilbert spaces, the notion of adjoint is generalised to the category setting. Dagger categories assign to each morphism f : A -> B an adjoint f* : B -> A. One can then consider f to be a morphism between A and B rather than going in any particular direction A -> B or B -> A. A cyclic operad is an operad with something akin to an adjoint; the action of the symmetric group interchanges the input objects and the output objects. This thesis contains a theory of quasi cyclic operads, including astroidal sets (presheaves over a category of unrooted trees) and a nerve theorem. Then, the shape of morphisms can be changed. Operads extend categories by allowing morphisms to have multiple inputs, while infinity operads extend infinity categories by being presheaves over the category of rooted trees rather than the category of paths. This can be further generalised to presheaves over other categories, in particular the category of graphs, to facilitate a connection with surfaces of higher genus. Graphical sets are the cyclic, higher genus analogue of simplicial sets. They are used to represent infinity modular operads. This thesis contains an exploration of each of the four models of infinity categories extended to infinity modular operads, with a focus on quasi modular operads and the inner Kan condition. Finally, given the applications in TQFTs and Grothendieck-Teichmuller theory, I construct an infinity modular operad of surfaces.
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ItemOn the existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric( [1982])In this thesis, we will prove that in the 3-dimensional sphere endowed with any Riemannian metric (denoted by N) there exists an embedded minimal 2-dimensional sphere. (From introduction)