Magister Scientiae - MSc (Mathematics and Applied Mathematics)
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Item Persistent homology and the application of topological data analysis to astronomical data(University of the Western Cape, 2024) Randall, JessicaThis thesis addresses a key challenge in topology: determining whether spaces are homeomorphic, which requires establishing a continuous, invertible mapping. Traditional methods based on basic invariants like connectivity and compactness often fall short for complex classifications. To address this, we draw on advanced concepts from algebraic topology, particularly the fundamental and homology groups introduced by Henri Poincar ́e and Enrico Betti, as well as Betti numbers that quantify the dimensions of “ holes ” in spaces. We analyze spatial structures through homology by focusing on missing elements. The p-th homology group Hp enables comparisons between the p-th cycle group Zp and the p-th boundary group Bp, allowing us to isolate sig- nificant topological features.Item A mathematical model for managing equity-linked pensions(University of the Western Cape, 2006) Julie, ElmeriePension fund companies manage and invest large amounts of money on behalf of their members. In return for their contributions, members expect a benefit at termination of their contract. Due to the volatile nature of returns that pension funds attain, pension companies started attaching a minimum guaranteed amount to member's benefits. In this mini-thesis we look at the pioneering work of Brennan and Schwartz [10] for pricing these minimum guarantees. The model they developed prices these minimum guarantees using option pricing theory. We also look at the model proposed by Deelstra et al. [13] which prices minimum guarantees in stochastic financial setting. We conclude this mini-thesis with new contributions where we look at simple alternative ways of pricing minimum guarantees. We conclude this mini-thesis with an approach, related to the work of Brenan and Schwartz [10], whereby the member's benefit is maximised for a given minimum guaranteed amount, which comprises of multi-period guarantees. We formulate a method to find the optimal stream of these multi-period guarantees.