Masters Degrees (Mathematical Sciences)

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Browsing Masters Degrees (Mathematical Sciences) by browse.metadata.advisor "Bartlett, Bruce"

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    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.
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    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.