Department of Mathematical Sciences

Permanent URI for this community

https://scholar.sun.ac.za/handle/10019.1/223

Browse

Browsing Department of Mathematical Sciences by browse.metadata.advisor "Bartlett, Bruce"

Now showing 1 - 5 of 5
  • Thumbnail Image
    Item
    Canonical connections in Riemannian and Hermitian geometry
    (Stellenbosch : Stellenbosch University, 2024-03) Sarah, Brian; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: This thesis presents explicit calculations of three naturally occurring connec- tions in Riemannian and Hermitian geometry. Namely, the Levi-Civita con- nection and the ambient connection in Riemannian geometry, and the Chern connection and the ambient connection in Hermitian geometry. Precisely, we show that the Chern connection and the ambient connection are equal on the tautological line bundle over CP¹. Next, we show that the Levi-Civita con- nection and the ambient connection are equal on the tangent bundle of the two-sphere. Finally, we compute the Chern connection on the tangent bundle of the two-sphere regarded as a Hermitian holomorphic line bundle and show that it is equal to the Levi-Civita connection on the tangent bundle of the two-sphere.
  • Thumbnail Image
    Item
    Coherent loop states and their applications in geometric quantization
    (Stellenbosch : Stellenbosch University, 2024-03) Nzaganya, Nzaganya Edson; Bartlett, Bruce; Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences.
    ENGLISH ABSTRACT: In the first part of this study, we study coherent loop states (also known as Bohr‑ Sommerfeld states) on 𝑆², with application to the representation theory of 𝑆𝑈(2). These states offer a precise bridge between the classical and quantum descriptions of angular momentum. We show that they recover the usual basis of angular mo‑ mentum eigenstates used in physics, and give a self‑contained proof of the asymp‑ totics of their inner products. As an application, we use these states to derive Little‑ john and Yu’s geometric formula for the asymptotics of the Wigner matrix elements. In the second part of this thesis, we consider coherent loop states on a general Riemann surface 𝑀. We show that for quasi‑regular polarizations of 𝑀, the second derivatives of the Bergman kernel on the diagonal of 𝑀 can be computed precisely in terms of the Kähler form of 𝑀. Therefore, the asymptotics of the inner product of coherent loop states can be computed using the complex stationary phase principle. This gives an alternative proof, for quasi‑regular polarized Riemann surfaces, of a variant of a result of Borthwick, Paul and Uribe.
  • Thumbnail Image
    Item
    Euler 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 [5] (which is based on a paper of Abrams [6]), 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.
  • Thumbnail Image
    Item
    Investigations 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.
  • 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.