Browsing by Author "Bavuma, Yanga"

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    Generalisations of the theorem of Peter and Weyl
    (2025) Stevenson, Elizabeth; Russo, Francesco G; Bavuma, Yanga
    Introducing some classical concepts in the representation theory of compact groups, we study the historical background of the Fourier Theorem and how this result was used to solve the heat equation. Then we sketch the main steps of the proof of the Fourier Theorem, explaining how it can be interpreted as a special case of a more general argument, which is used in the proof of the Peter-Weyl Theorem. A comparative analysis of different arguments, which have been used to prove the Fourier Theorem and the Peter-Weyl Theorem in the literature, is illustrated in the present thesis. Therefore, we note that some arguments apply to compact connected abelian groups while others can't be applied. Inspecting the details of the proofs, we observe that the use of the Stone-Weierstrass Theorem plays a fundamental role, so we discuss the relevance of the Stone-Weiestrass Theorem in the proof of Peter-Weyl Theorem.
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    Some combinatorial aspects in algebraic topology and geometric group theory
    (2018) Bavuma, Yanga
    The present Msc thesis deals with classical topics of topology and it has been written, referring to [C. Kosniowski, Introduction to Algebraic Topology, Cambridge University Press, 1980, Cambridge], which is a well known textbook of algebraic topology. It has been selected a list of main exercises from this reference, whose solutions were not directly available, or subject to differerent methods. In fact combinatorial methods have been preferred and the result is a self-contained dissertation on the theory of the fundamental group and of the coverings. Finally, there are some recent problems in geometric group theory which are related to the presence of finitely presented groups which appear naturally as fundamental groups.