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Item Limiting Configurations for the SU(1,2) Hitchin Equation(2022) Na, Xuesen; Wentworth, Richard A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation studies the SU(1,2) Higgs bundle and a limiting behavior of solutions of the SU(1,2) Hitchin's self-duality equation. On a closed Riemann surface $X$ of genus $g\ge 2$, an SU(1,2) Higgs bundle consists of the following data: a rank two holomorphic vector bundle $F$ and the holomorphic maps $\beta: L\otimes K_X^{-1}\to F$, $\gamma: F\to L\otimes K_X$ where $L=\det F^\ast$. The Hitchin map of the moduli space of SU(1,2) Higgs bundles takes $(F,\beta,\gamma)$ to the quadratic differential $q=\gamma\circ\beta$. For an SU(1,2) Higgs bundle $(F,\beta,\gamma)$, the Hitchin equation is a non-linear PDE of hermitian metric $h$ on $F$. The existence of a unique solution follows from the stability condition. For a stable SU(1,2) Higgs bundle $(F,\beta,\gamma)$, we give an explicit description of the behavior of $h_t$, the unique solution of SU(1,2) Hitchin equation for the family $(F,t\beta,t\gamma)$ in the case where $q$ has simple zeros in the limit $t\to\infty$. In Chapter 1, we review the notion of $G$ Higgs bundles and focus on the case $G=$SU(1,2). The simple zeros of $q=\gamma\circ\beta$ are one of the three types: (1) a zero of $\beta$, (2) a zero of $\gamma$, or (3) neither. We present a stability condition in terms of the number of zeros of each type. We also review notions of the filtered bundle and the wild harmonic bundle. In Chapter 2, we give an explicit description of the fiber of the Hitchin map in terms of a fiber bundle over the Jacobian of $X$ with unirational fibers. The fiber is a GIT quotient of a $\mathbb{C}^\times$-action on $(\mathbb{P}^1)^{4g-4}$. The base parametrizes the choice of a line bundle $L$. The fiber gives parameters for a Hecke modification $\iota: F\to V$ which realizes $F$ as a rank-two locally free subsheaf of $V=L^{-2}K_X\oplus LK_X$. We show that the stable locus is a coarse moduli space of the appropriate moduli functor. In Chapters 3 and 4, we study the Hitchin equation for the family $(F,t\beta,t\gamma)$ as $t\to\infty$. In particular, we show that the limiting configuration $h_\infty$ satisfies the decoupled Hitchin equation and is induced from a harmonic metric $h_L$ on $L$ via the Hecke modification $\iota: F\to V$. The metric $h_L$ is adapted to a filtered line bundle $(L,\underline{\lambda_\infty})$ where the weights $\underline{\lambda_\infty}$ are specified by a rule depending on the types of zeros and their count. We prove the convergence of $h_t$ to $h_\infty$ after appropriate normalizing by gluing local model solutions constructed from wild harmonic bundles on $\mathbb{P}^1$ over disks around the zeros to a solution of the decoupled equation on the complement.Item Parabolic Higgs bundles and the Deligne-Simpson Problem for loxodromic conjugacy classes in PU(n,1)(2017) Maschal Jr, Robert Allan; Wentworth, Richard; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis we study the Deligne-Simpson problem of finding matrices $A_j\in C_j$ such that $A_1A_2\ldots A_k = I$ for $k\geq 3$ fixed loxodromic conjugacy classes $C_1,\ldots,C_k$ in $PU(n,1)$. Solutions to this problem are equivalent to representations of the $k$ punctured sphere into $PU(n,1)$, where the monodromy around the punctures are in the $C_j$. By Simpson's correspondence \cite{s1}, irreducible such representations correspond to stable parabolic $U(n,1)$-Higgs bundles of parabolic degree 0. A parabolic $U(n,1)$-Higgs bundle can be decomposed into a parabolic $U(1,1)$-Higgs bundle and a $U(n-1)$ bundle by quotienting out by the rank $n-1$ kernel of the Higgs field. In the case that the $U(1,1)$-Higgs bundle is of loxodromic type, this construction can be reversed, with the added consequence that the stability conditions of the resulting $U(n,1)$-Higgs bundle are determined only by the kernel of $\Phi$, the number of marked points, and the degree of the $U(1,1)$-Higgs bundle. With this result, we prove our main theorem, which says that when the log eigenvalues of lifts $\widetilde{C}_j$ of the $C_j$ to $U(n,1)$ satisfy the inequalities in \cite{biswas} for the existence of a stable parabolic bundle, then there is a stable parabolic $U(n,1)$-Higgs bundle whose monodromies around the marked points are in $\widetilde{C}_j$. This new approach using Higgs bundle techniques generalizes the result of Falbel and Wentworth in \cite{fw1} for fixed loxodromic conjugacy classes in $PU(2,1)$. This new result gives sufficient, but not necessary, conditions for the existence of an irreducible solution to the Deligne-Simpson problem for fixed loxodromic conjugacy classes in $PU(n,1)$. The stability assumption cannot be dropped from our proof since no universal characterization of unstable bundles exists. In the last chapter, we explore what happens when we change the weights of the stable kernel in the special case of three fixed loxodromic conjugacy classes in $PU(3,1)$. Using the techniques from \cite{fw2}, \cite{fw1}, and \cite{paupert}, we can show that our construction implies the existence of many other solutions to the problem.