Thesis advisor: Martin Bridgeman / We begin this dissertation by studying the relationship between the Poincaré metric of a simply connected domain Ω ⊂ ℂ and the geometry of Dome(Ω), the boundary of the convex hull of its complement. Sullivan showed that there is a universal constant K[subscript]eq[subscript] such that one may find a conformally natural K[subscript]eq[subscript]-quasiconformal map from Ω to Dome(Ω) which extends to the identity on ∂Ω. Explicit upper and lower bounds on K[subscript]eq[subscript] have been obtained by Epstein, Marden, Markovic and Bishop. We improve upon these upper bounds by showing that one may choose K[subscript]eq[subscript] ≤ 7.1695. As part of this work, we provide stronger criteria for embeddedness of pleated planes. In addition, for Kleinian groups Γ where N = ℍ³/Γ has incompressible boundary, we give improved bounds for the average bending on the convex core of N and the Lipschitz constant for the homotopy inverse of the nearest point retraction. In the second part of this dissertation, we prove an extension of Basmajian's identity to n-Hitchin representations of compact bordered surfaces. For 3-Hitchin representations, we provide a geometric interpretation of this identity analogous to Basmajian's original result. As part of our proof, we demonstrate that for a closed surface, the Lebesgue measure on the Frenet curve of an n-Hitchin representation is zero on the limit set of any incompressible subsurface. This generalizes a classical result in hyperbolic geometry. In our final chapter, we prove the Bridgeman-Kahn identity for all finite volume hyperbolic n-manifolds with totally geodesic boundary. As part of this work, we correct a commonly referenced expression of the volume form on the unit tangent bundle of ℍⁿ in terms of the geodesic end point parametrization. / Thesis (PhD) — Boston College, 2016. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

We develop a new geometric method of understanding principal G-Higgs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p), SU(p,p), and Sp(2p,2p). Further, we give some applications of our results, and discuss open questions.

A revised lithostratigraphy and glacial history of north Hertfordshire and south Bedfordshire is based upon detailed textural data in the clay to fine gravel fraction, carbonate content, small clast lithological data and macrofabrics, derived from laboratory and field analyses of tills from 30 sites. These include four deep boreholes sunk within the Hitchin Gap. A range of statistical procedures was used, including multivariate analysis of the petrographic properties, enabling the identification of tills from two separate incursions into the Gap. A further till-type was identified in south Bedfordshire indicating an ice advance from the northwest/NNW extending at least as far east as Milton Bryan. Statistical comparison with tills in the neighbouring Vale of St. Albans suggested the presence of the Ware Member till within the Gap. Two hypotheses are suggested to explain variations in lithological content of tills north of the Chalk scarp. The first envisages ice entering the study area along the different trajectories suggested by Fish and Whiteman (2001). During the early part of the glaciation, ice reaching the west of the study area would approach from the north, crossing a shorter distance over Chalk bedrock and collecting less chalk and flint than ice moving into the eastern part of the study area. The second hypothesis invokes an incursion of ice from a northwest - NNW direction into the west of the study area, depositing a chalk-free till. This is subsequently assimilated by ice from the northeast, resulting in the final deposition of a homogeneous mixture of debris from the two advances, with a lower chalk content than tills found to the east. The outcome of either of these scenarios is a till with a low acid-soluble content and low flint/quartz ratio in the west of the study area, as found during this work. Within the Hitchin Gap, a lobe of ice, probably an early part of the northeasterly advance, deposited a lower till. This is considered to be earlier than the Ware Member till and has more variable lithological characteristics and a finer matrix that the higher tills. The latter are mainly melt-out, flow or slumped tills with occasional instances of lodgement and deformation. They represent in situ wasting of dead ice within the Gap. Surface tills in the Gap form a continuum with tills to the north and comprise mainly deformation tills, deposited by the final northeasterly re-advance of ice responsible for widespread coverage of the region, with the exception of the Chiltern Hills southwest of Hitchin. No evidence is found of more than one lithostratigraphic unit of till outside the Hitchin Gap.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O objetivo desse trabalho à fornecer uma demonstraÃao para as desigualdades de Hitchin-Thorpe e Miyaoka-Yau. Inicialmente forneceremos uma decomposiÃÃo ortogonal para o tensor curvatura, em seguida mostraremos como o operador curvatura pode ser definido a partir do tensor curvatura. Com o intuito de cumprir o objetivo proposto, iremos provar o Teorema de Gauss-Bonnet em dimensÃo 4, para isso utilizaremos um resultado devido a Allendoerfer e forneceremos uma fÃrmula integral para o cÃlculo da caracterÃstica de Euler de uma variedade Riemanniana de dimensÃo 4. AlÃm disso, definiremos o conceito de assinatura em uma variedade Riemanniana e exibiremos uma fÃrmula integral para a obtenÃÃo deste objeto, para isso utilizaremos o Teorema de Assinatura de Hirzebruch em dimensÃo 4 e pouco da Teoria de Chern-Weil que nos fornece uma conexÃo entre a topologia algÃbrica e a geometria diferencial. Por fim, mostraremos como as fÃrmulas que foram obtidas podem ser utilizadas na demonstraÃao das desigualdades citadas inicialmente. / The aim of this work is to present a proof of the Hitchin-Thorpe and Miyaoka-Yau inequalities. First we provide an orthogonal decomposition for the curvature tensor, and then we show how the curvature operator can be defined from the curvature tensor. In order to fulfill the proposed objective, we prove the Gauss-Bonnet Theorem in dimension 4, to do this we use a result due Allendoerfer and we present an integral formula for the Euler characteristic computation on a Riemannian 4-manifold. Furthermore, we define the concept of signature in a Riemannian manifold e we exhibit an integral formula for the achievement of this object, for this we use the Hirzebruch Signature Theorem in di- mension 4 and the Chern-Weil Theory which provides us a connection between algebraic topology and differential geometry. Finally, we show how the earlier formulas can be used in the demonstration of the initial inequalities.

[pt] O espaço de polígonos de um grupo de Lie é definido como a redução simplética em um produto de órbitas pela ação coadjunta. Neste trabalho comparamos alguns sistemas integráveis definidos em espaços de módulos de polígonos, matrizes e fibrados, tais como o sistema de Kapovich–Millson, o modelo de Gaudin e a aplicação de Hitchin. / [en] The Polygon Space of a Lie group is defined as the symplectic reduction of a product of orbits by the coadjoint action. In this work we compare integrable systems defined on different moduli spaces of polygons, matrices and bundles, such as Kapovich–Millson s system, Gaudin s model and the Hitchin s map.

Ce travail introduit la notion de schéma d'Atiyah et Hitchin. Une variété algébrique raisonnable Y étant fixée, il s'agit d'une famille de nouveaux schémas, indexée par un entier positif m et notée $R_m(Y)$. Nous étudions les propriétés homotopiques de ces « espaces » au sens de Morel et Voevodsky. Les schémas $F_m$ des fractions rationnelles pointées de degré m constituent un exemple fondateur et fondamental. Du point de vue topologique, les travaux de G. Segal et F. Cohen et al. montrent que l'espace $F_m(C)$ approxime l'espace de lacets $Ω^2 S^3$. Nous formulons une série précise de conjectures visant à généraliser ces résultats dans un cadre algébrique. Le schéma $R_m(Y)$ approximerait l'espace de lacets motivique $Ω^{P¹} Σ^{P¹} Y$. Nous obtenons plusieurs résultats dans cette direction. En particulier : 1) Nous déterminons l'ensemble des composantes connexes algébriques naïves du schéma de fractions rationnelles $F_m$, au-dessus d'un corps de base. Le calcul est simple et élémentaire. On retrouve, à une complétion près, le groupe des classes d'homotopie d'endomorphismes pointés de la droite projective $P¹$, tel que calculé par Morel. 2) Nous construisons un morphisme algébrique reliant $R_mY$ à $Ω^{P¹} Σ^{P¹} Y$. 3) Lorsque Y est une variété algébrique complexe, nous explicitons le type d'homotopie de l'espace topologique $R_m(Y)(C)$ comme un foncteur en $Y(C)$. De plus, nous montrons que l'espace $R_m(Y)(C)$ admet un scindement stable dont les facteurs sont ceux du scindement de Snaith de l'espace $Ω² Σ² Y (C)$.

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.

Soit X une courbe projective lisse et irréductible munie d'une involution σ. Dans cette thèse, nous étudions les fibrés vectoriels invariants and anti-invariants sur X sous l'action induite par σ. On introduit la notion de modules σ-quadratiques et on l'utilise, avec GIT, pour construire ces espaces de modules, puis on en étudie certaines propriétés. Ces espaces de modules correspondent aux espaces de modules de G-torseurs parahoriques sur la courbe X/σ , pour certains schémas en groupes parahoriques G de type Bruhat-Tits, qui sont twistés dans le cas des anti-invariants. Nous développons les systèmes de Hitchin sur ces espaces de modules et on les utilise pour dériver une classification de leurs composantes connexes en les dominant par des variétés de Prym. On étudie aussi le fibré déterminant sur les espaces de modules des fibrés vectoriels anti-invariants. Dans certains cas, ce fibré en droites admet certaines racines carrées appelées fibrés Pfaffiens. On montre que les espaces des sections globales des puissances de ces fibrés en droites (les espaces des fonctions thêta généralisées) peuvent être canoniquement identifier avec les blocs conformes associés aux algèbres de Kac-Moody affines twistées de type A(2). / Let X be a smooth irreducible projective curve with an involution σ. In this dissertation, we studythe moduli spaces of invariant and anti-invariant vector bundles over X under the induced action of σ. We introduce the notion of σ-quadratic modules and use it, with GIT, to construct these moduli spaces, and than we study some of their main properties. It turn out that these moduli spaces correspond to moduli spaces of parahoric G-torsors on the quotient curve X/σ, for some parahoric Bruhat-Tits group schemes G, which are twisted in the anti-invariant case.We study the Hitchin system over these moduli spaces and use it to derive a classification of theirconnected components using dominant maps from Prym varieties. We also study the determinant of cohomology line bundle on the moduli spaces of anti-invariant vector bundles. In some cases this line bundle admits some square roots called Pfaffian of cohomology line bundles. We prove that the spaces of global sections of the powers of these line bundles (spaces of generalized theta functions) can be canonically identified with the conformal blocks for some twisted affine Kac-Moody Lie algebras of type A(2).

Ce travail de thèse a pour but de construire et d’étudier une fibration de Hitchin pour les groupes qui apparaît naturellement lorsque l’on essaie de géométriser la formule des traces. On commence par construire une telle fibration en utilisant le semi-groupe de Vinberg. Sur ce semi-groupe de Vinberg, on montre qu’il existe un certain morphisme « polynôme caractéristique » muni d’une section naturelle, de même que dans le cas des algèbres de Lie. On montre également que l’on peut construire un centralisateur régulier au-dessus de cette base des polynômes caractéristiques qui est un schéma en groupes commutatif et lisse.On s’intéresse alors à des variantes pour les groupes des fibres de Springer affines pour lesquelles on remarque que l’introduction du semi-groupe de Vinberg permet d’obtenir une condition d’intégralité analogue à celle de Kazhdan-Lusztig. Ces fibres de Springer affines sont des analogues locaux des fibres de Hitchin. On obtient alors une formule de dimension pour ces fibres.Dans un troisième temps, on s’intéresse à l’aspect global de cette fibration pour laquelle on donne une interprétation modulaire et sur laquelle on construit l’action d’un champ de Picard, issu du centralisateur régulier. L’espace total de cette fibration étant en général singulier, nous étudions son complexe d’intersection. Cet espace de Hitchin s’obtient naturellement comme l’intersection du champ de Hecke avec la diagonale du champ des G-torseurs et on démontre que sur un ouvert suffisamment gros de la base de Hitchin, le complexe d’intersection de l’espace de Hitchin s’obtient par restriction de celui du champ de Hecke corrrespondant.Enfin, dans la dernière partie de cette thèse, on établit un théorème du support dans le cas où l’espace total est singulier analogue à celui de Ngô et l’on démontre que, dans le cas de la fibration de Hitchin, les supports qui interviennent sont reliés aux strates endoscopiques. / This main goal of this work is to construct and study the properties of Hitchin fibration for groups which appears naturally when we try to geometrize the trace formula. We begin by constructing this fibration using the Vinberg’s semigroup. On this semigroup, we show that there exists a characteristic polynomial morphism equipped with a natural section, analog at the Kostant’s one in the case of Lie algebras. We also show that there exists on the base of characteristic polynomials a regular centralizer scheme, which is a smooth commutative group scheme.Then, we are interested in some variant of affine Springer fibers, for which we see that the Vinberg’s semigroup appears naturally to obtain an integrality condition analog to Kazhdan-Lusztig’s one. These affine Springer fibers are local incarnation of Hitchin fibers.In a third time, we go back to the global case and give a modular interpretation of this new Hitchin fibration on which we construct an action of a Picard stack, coming from the regular centralizer.The total space of this fibration, even on the generically regular semisimple locus will be singular and we want to understand his intersection complex. This space can be obtained as the intersection of the Hecke stack with the diagonal of the stack of G-bundles and we show that on a sufficiently big open subset of the Hitchin base, the intersection complex of the Hitchin’s space is the restriction of the corresponding intersection complex on the Hecke stack.Finally, in the last part of this work, we establish a support theorem in the case of a singular total space, generalizing Ngo’s theorem et we show that in the case of Hitchin fibration, the supports that appear are related to the endoscopic strata.

Soit $M$ une variété projective lisse. Soit $\mathscr{F}$ une filtration holomorphe sur $M$, c'est à dire une filtration d'un fibré vectoriel holomorphe $\mathcal{F}$ induite par des sous-fibrés. Nous introduisons une notion de Gieseker stabilité pour de tels objets puis donnons une condition analytique équivalente en terme de métriques sur $\mathcal{F}$, dites équilibrées au sens de S.K. Donaldson, provenant d'une construction de la Théorie des Invariants Géométriques. Si le fibré $\mathcal{F}$ peut être muni d'une métrique $h$ solution de l'équation $\boldsymbol{\tau}$-Hermite-Einstein étudiée par \'lvarez-C\'{o}nsul et Garc\'a-Prada:<br />$$\sqrt\Lambda F_h = \sum_i \widetilde_i\pi^_$$<br />alors nous prouvons que la suite de métriques équilibrées existe, converge et sa limite est, à un changement conforme, solution de l'équation précédente. De ce résultat nous déduisons, par réduction dimensionnelle, un théorème d'approximation dans le cas des équations Vortex de Bradlow ainsi que leurs généralisations aux équations couplées Vortex.