Browsing Department of Mathematical Sciences by browse.metadata.advisor "Bartlett, Bruce"
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- ItemEuler Classes and Frobenius Algebras(Stellenbosch : Stellenbosch University, 2019-04) Ranaivomanana, Valimbavaka Hosana; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics.ENGLISH ABSTRACT : This thesis investigates the relationship between the handle element of the De Rham cohomology algebra of a compact oriented smooth manifold, thought of as a Frobenius algebra, and the Euler class of the manifold. In this way it gives a complete answer to an exercise posed in the monograph of Kock  (which is based on a paper of Abrams ), where one is asked to show that these two classes are equal. Firstly, an overview of De Rham cohomology, Thom and Euler classes of smooth manifolds, Poincaré duality, Frobenius algebras, and their graphical calculus is given. Finally, it is shown that the handle element and the Euler class are indeed equal for even-dimensional manifolds. However, they are not equal for odd-dimensional manifolds.
- ItemInvestigations on the Wigner derivative and on an integral formula for the quantum 6j symbols(Stellenbosch : Stellenbosch University, 2022-04) Ranaivomanana, Valimbavaka Hosana; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.ENGLISH SUMMARY: wo separate studies are done in this thesis: 1. TheWigner derivative is the partial derivative of dihedral angle with respect to opposite edge length in a tetrahedron, all other edge lengths remaining fixed. We compute the inverse Wigner derivative for spherical tetrahedra, namely the partial derivative of edge length with respect to opposite dihedral angle, all other dihedral angles remaining fixed. We show that the inverse Wigner derivative is actually equal to theWigner derivative. 2. We investigate a conjectural integral formula for the quantum 6j symbols suggested by Bruce Bartlett. For that we consider the asymptotics of the integral and compare it with the known formula for the asymptotics of the quantum 6j symbols due to Taylor and Woodward. Taylor and Woodward’s formula can be rewritten as a sum of two quantities: ins and bound. The asymptotics of the integral splits into an interior and boundary contribution. We successfully compute the interior contribution using the stationary phase method. The result is indeed quite similar to although not exactly the same as ins. Though we expect the boundary contribution to be similar to bound, the computation is left for future work.
- ItemOriented 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.