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 Title: Gluing constructions for Higgs bundles over a complex connected sum Author(s): Kydonakis, Georgios A. Director of Research: Bradlow, Steven B Doctoral Committee Chair(s): Nevins, Thomas Doctoral Committee Member(s): Albin, Pierre; Dunfield, Nathan M Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Higgs bundles, character variety, topological invariants Abstract: For a compact Riemann surface of genus $g\ge 2$, the components of the moduli space of $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Higgs bundles, or equivalently the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-character variety, are partially labeled by an integer $d$ known as the Toledo invariant. The subspace for which this integer attains a maximum has been shown to have $3\cdot {{2}^{2g}}+2g-4$ many components. A gluing construction between parabolic Higgs bundles over a connected sum of Riemann surfaces provides model Higgs bundles in a subfamily of particular significance. This construction is formulated in terms of solutions to the Hitchin equations, using the linearization of a relevant elliptic operator. Issue Date: 2018-04-02 Type: Thesis URI: http://hdl.handle.net/2142/100920 Rights Information: Copyright 2018 Georgios A. Kydonakis Date Available in IDEALS: 2018-09-04 Date Deposited: 2018-05
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