Undergraduate Research

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https://hdl.handle.net/1920/12807
Honors Theses, papers, posters, and other works from undergraduate students in the Department of Mathematical Sciences

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    Lamplighters Groups: Their Presentations, Caylely Graphs, and Generating Automata
    (2023) Miller, Holly
    The focus of this paper is the traditional lamplighter group L2 = (Z/2Z) wrZ and the broader class of lamplighter groups LG = Gwr Z, where G is a finite group. These groups have seen direct use in papers such as [7], in which Grigorchuk, Linnell, Schick, and Z˙ uk show there is a closed Riemannian manifold (with the HNN-extension of L2 as its fundamental group) that has a rational L2-Betti number. Other results rely on generalizing or building atop the lamplighter group, as is done by Brieussel and Zheng in [3] to expand the known possible speed, entropy, isoperimetric profile, return probability, and compression gap attainable by random walks on finitely generated groups. This paper provides the algebraic background needed to understand the definition of the traditional lamplighter group, as well as its presentation, Caylely graph, and representation as an automaton group. All three of these constructions provide distinct avenues to generalize the lamplighter group. The sort of generalization discussed in this paper—moving from L2 to LG—is quite natural from the perspective of the lamplighter group’s Cayley graph and presentation. The case of finding a suitable automaton to generate such generalizations is more interesting; only some lamplighter groups LG can be generated by automata. When G is a finite Abelian group, then LG can be generated by the Cayley machine C(G) of G, and when G is a finite Non-Abelian group there is no automaton which can generate the group. These Cayley machines are interesting in their own right, so we explore the more general structure of the groups generated by Cayley machines.
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    The Persistence of Data: A Road Map
    (2022-04-22) Pothagoni, Shrunal
    The purpose of data mining is to use advanced mathematical and statistical techniques to extract quantitative information from large data sets. These tools are incredibly powerful and in conjunction with machine learning algorithms allow for extremely accurate pattern prediction. However, there are various datasets that have qualitative properties that cannot be discerned using classic data mining techniques. Topological Data Analysis (TDA) is a field developed within the last two decades that uses methods in topology to extract such qualitative features. In this paper we will study how to use abstract simplicial complexes on point cloud data sets to find their most ‘optimal’ topology using computational homology.
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    Vertex operator algebras: Finite-dimensional cases and conformal blocks
    (2022-04-29) Andrews, George
    Vertex operator algebras are algebraic objects analogous to both commutative associative algebras with identity and Lie algebras. They provide a way of rigorously constructing a particular family of quantum field theories called rational conformal field theories. In this thesis we construct the simplest class of examples of vertex operator algebras, namely the finite dimensional ones, and prove basic results on modules of these vertex operator algebras and spaces of conformal blocks associated to smooth projective curves. We also construct the vertex operator algebra associated with the $\mathfrak{sl}_2(\mathbb{C})$ WZW model in the non-critical case. When combined with the FRS theorem for rational conformal field theories, vertex operator algebra theory can be used to rigorously construct one of the simplest examples of holographic duality: the Chern-Simons-WZW model correspondence.