Browsing by Author "Goosen, Gerrit"

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Oriented 123-TQFTs via String-Nets and State-Sums
    (Stellenbosch : Stellenbosch University, 2018-03) Goosen, Gerrit; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT :In a series of papers Bartlett, Douglas, Schommer-Pries, and Vicary discovered a finite generators-and-relations presentation of the oriented bordism bicategory. This simplifies the task of finding oriented once-extended topological quantum field theories (123-TQFTs). We combine this result with the theory of string-nets, specifically Kirillov’s generalization of the theory to surfaces with boundary, and construct an oriented 123-TQFT based on string-nets. Previously, string-nets were only understood at the level of surfaces - in particular, it was not known how to assign linear maps to cobordisms which changed the topology of the surface. We also reformulate the extended Turaev-Viro theory developed by Balsam and Kirillov into the bicategorical generators-and-relations formalism, and use this to prove that the string-net and Turaev-Viro 123-TQFTs are equivalent.
  • Thumbnail Image
    Item
    Relational representations for bounded lattices with operators
    (Stellenbosch : University of Stellenbosch, 2010-03) Goosen, Gerrit; Rewitzky, Ingrid; University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: Within lattice theory, an interesting question asked is whether a given abstract lattice may be represented concretely as subsets of a closure system on a topological space. This is true for boolean algebras, bounded distributive lattices and arbitrary bounded lattices. In particular, there are a multitude of ways to represent bounded lattices. We present some of these ideas, as well as an analysis of the differences between them. We further investigate the attempts that were made to extend the above representations to lattices endowed with operators, in particular the work done on bounded distributive lattices with operators. We then make a new contribution by extending this work to arbitrary bounded lattices with operators. We also show that the so-called sufficiency operator has a relational representation in the bounded lattice case.