@phdthesis{Kraus2003,
  author    = {Kraus, Daniela},
  title     = {Conformal pseudo-metrics and some applications},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-9193},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2003},
  abstract  = {The point of departure for the present work has been the following free boundary value problem for analytic functions \$f\$ which are defined on a domain \$G \subset \mathbb{C}\$ and map into the unit disk \$\mathbb{D}= \{z \in \mathbb{C} : |z|<1 \}\$. Problem 1: Let \$z_1, \ldots, z_n\$ be finitely many points in a bounded simply connected domain \$G \subset \mathbb{C}\$. Show that there exists a holomorphic function \$f:G \to \mathbb{D}\$ with critical points \$z_j\$ (counted with multiplicities) and no others such that \$\lim_{z \to \xi} \frac{|f'(z)|}{1-|f(z)|^2}=1\$ for all \$\xi \in \partial G\$. If \$G=\mathbb{D}\$, Problem 1 was solved by K?nau [5] in the case of one critical point, and for more than one critical point by Fournier and Ruscheweyh [3]. The method employed by K?nau, Fournier and Ruscheweyh easily extends to more general domains \$G\$, say bounded by a Dini-smooth Jordan curve, but does not work for arbitrary bounded simply connected domains. In this paper we present a new approach to Problem 1, which shows that this boundary value problem is not an isolated question in complex analysis, but is intimately connected to a number of basic open problems in conformal geometry and non-linear PDE. One of our results is a solution to Problem 1 for arbitrary simply connected domains. However, we shall see that our approach has also some other ramifications, for instance to a well-known problem due to Rellich and Wittich in PDE. Roughly speaking, this paper is broken down into two parts. In a first step we construct a conformal metric in a bounded regular domain \$G\subset \mathbb{C}\$ with prescribed non-positive Gaussian curvature \$k(z)\$ and prescribed singularities by solving the first boundary value problem for the Gaussian curvature equation \$\Delta u =-k(z) e^{2u}\$ in \$G\$ with prescribed singularities and continuous boundary data. This is related to the Berger-Nirenberg problem in Riemannian geometry, the question which functions on a surface R can arise as the Gaussian curvature of a Riemannian metric on R. The special case, where \$k(z)=-4\$ and the domain \$G\$ is bounded by finitely many analytic Jordan curves was treated by Heins [4]. In a second step we show every conformal pseudo-metric on a simply connected domain \$G\subseteq \mathbb{C}\$ with constant negative Gaussian curvature and isolated zeros of integer order is the pullback of the hyperbolic metric on \$\mathbb{D}\$ under an analytic map \$f:G \to \mathbb{D}\$. This extends a theorem of Liouville which deals with the case that the pseudo-metric has no zeros at all. These two steps together allow a complete solution of Problem 1. Contents: Chapter I contains the statement of the main results and connects them with some old and new problems in complex analysis, conformal geometry and PDE: the Uniformization Theorem for Riemann surfaces, the problem of Schwarz-Picard, the Berger-Nirenberg problem, Wittich's problem, etc.. Chapter II and III have preparatory character. In Chapter II we recall some basic results about ordinary differential equations in the complex plane. In our presentation we follow Laine [6], but we have reorganized the material and present a self-contained account of the basic features of Riccati, Schwarzian and second order differential equations. In Chapter III we discuss the first boundary value problem for the Poisson equation. We shall need to consider this problem in the most general situation, which does not seem to be covered in a satisfactory way in the existing literature, see [1,2]. In Chapter IV we turn to a discussion of conformal pseudo-metrics in planar domains. We focus on conformal metrics with prescribed singularities and prescribed non-positive Gaussian curvature. We shall establish the existence of such metrics, that is, we solve the corresponding Gaussian curvature equation by making use of the results of Chapter III. In Chapter V we show that every constantly curved pseudo-metric can be represented as the pullback of either the hyperbolic, the euclidean or the spherical metric under an analytic map. This is proved by using the results of Chapter II. Finally we give in Chapter VI some applications of our results. [1,2] Courant, H., Hilbert, D., Methoden der Mathematischen Physik, Erster/ Zweiter Band, Springer-Verlag, Berlin, 1931/1937. [3] Fournier, R., Ruscheweyh, St., Free boundary value problems for analytic functions in the closed unit disk, Proc. Amer. Math. Soc. (1999), 127 no. 11, 3287-3294. [4] Heins, M., On a class of conformal metrics, Nagoya Math. J. (1962), 21, 1-60. [5] K?nau, R., L?gentreue Randverzerrung bei analytischer Abbildung in hyperbolischer und sph?ischer Geometrie, Mitt. Math. Sem. Giessen (1997), 229, 45-53. [6] Laine, I., Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin - New York, 1993.},
  subject      = {Freies Randwertproblem},
  language  = {en}
}
@phdthesis{Nagel2003,
  author    = {Nagel, Christian},
  title     = {Gl{\"a}ttungsverfahren f{\"u}r semidefinite Programme},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-8099},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2003},
  abstract  = {In dieser Arbeit werden Algorithmen zur L{\"o}sung von linearen semidefiniten Programmen beschrieben. Unter einer geeigneten Regularit{\"a}tsvoraussetzung ist ein semidefinites Programm {\"a}quivalent zu seinen Optimalit{\"a}tsbedingungen. Die Optimalit{\"a}tsbedingungen bzw. die Zentralen-Pfad-Bedingungen {\"u}berf{\"u}hren wir zun{\"a}chst durch matrixwertige NCP-Funktionen in ein nichtlineares Gleichungssystem. Dieses nichtlineare und teilweise nicht differenzierbare Gleichungssystem l{\"o}sen wir dann mit einem Newton-{\"a}hnlichen Verfahren. Durch die Umformulierung in ein nichtlineares Gleichungssystem muss w{\"a}hrend der Iteration nicht mehr explizit die positive (Semi-)Definitheit der beteiligten Matrizen beachtet werden. Weiter wird gezeigt, dass dieser Ansatz im Gegensatz zu Inneren-Punkte-Methoden sofort symmetrische Suchrichtungen erzeugt. Um globale Konvergenz zu erhalten, werden verschiedene Globalisierungsstrategien (Schrittweitenbestimmung, Trust-Region-Ansatz) untersucht. F{\"u}r das betrachtete Pr{\"a}diktor-Korrektor-Verfahren und das Trust-Region-Verfahren wird lokal superlineare Konvergenz unter strikter Komplementarit{\"a}t und Nichtdegeneriertheit gezeigt. Die theoretische Untersuchung eines nichtglatten Newton-Verfahrens liefert ein lokal quadratisches Konvergenzverhalten ohne strikte Komplementarit{\"a}t, wenn die Nichtdegeneriertheitsvoraussetzung geeignet modifiziert wird.},
  subject      = {Semidefinite Optimierung},
  language  = {de}
}
@phdthesis{Fleischmann2003,
  author    = {Fleischmann, Peter},
  title     = {Bildung reiner H{\"u}llen in vollst{\"a}ndig zerlegbaren torsionsfreien abelschen Gruppen},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-5979},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2003},
  abstract  = {Reine Untergruppen von vollst{\"a}ndig zerlegbaren torsionsfreien abelschen Gruppen werden Butlergruppen genannt. Eine solche Gruppe l{\"a}ßt sich als endliche Summe von rationalen Rang-1-Gruppen darstellen. Eine solche Darstellung ist nicht eindeutig. Daher werden Methoden entwickelt, die zu einer Darstellung mit reinen Summanden f{\"u}hren. Weiter kann aus dieser Darstellung sowohl die kritische Typenmenge als auch die Typuntergruppen direkt abgelesen werden. Dies vereinfacht die Behandlung von Butlergruppen mit dem Computer und gestattet dar{\"u}berhinaus eine elegantere Darstellung.},
  subject      = {Torsionsfreie Abelsche Gruppe},
  language  = {de}
}
@phdthesis{Preiss2002,
  author    = {Preiß, Martin},
  title     = {Analytizit{\"a}tseigenschaften gewichteter zentraler Pfade bei monotonen Komplementarit{\"a}tsproblemen und ihre Ausnutzung},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3969},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2002},
  abstract  = {Die vorliegende Arbeit untersucht die Analytizit{\"a}tseigenschaften unzul{\"a}ssiger Innerer-Punkte Pfade bei monotonen Komplementarit{\"a}tsproblemen und diskutiert m{\"o}gliche algorithmische Anwendungen. In Kapitel 2 werden einige matrixanalytische Konzepte und Resultate zusammengestellt, die f{\"u}r die Beweisf{\"u}hrung in den folgenden Kapiteln ben{\"o}tigt werden. Kapitel 3 gibt eine genaue Definition der Begriffe "monotones lineares Komplementarit{\"a}tsproblem" (LCP) bzw. "semidefinites monotones lineares Komplementarit{\"a}tsproblem" (SDLCP) und zeigt die Grundidee hinter den Innere-Punkte-Verfahren zur L{\"o}sung solcher Probleme. Kapitel 4 beinhaltet die analytischen Hauptresultate f{\"u}r monotone Komplementarit{\"a}tsprobleme. In Abschnitt 4.1 werden einige wohlbekannte Resultate {\"u}ber die Analytizit{\"a}tseigenschaften unzul{\"a}ssiger Innerer-Punkte-Pfade f{\"u}r LCP's wiedergegeben. Diese werden in Abschnitt 4.2 auf den semidefiniten Fall {\"u}bertragen. Unter der Annahme, dass das zugrundeliegende SDLCP eine strikt komplement{\"a}re L{\"o}sung besitzt, wird gezeigt, dass die Inneren-Punkte-Pfade sogar noch im Randpunkt analytisch sind. Kapitel 5 benutzt die Resultate aus Kapitel 4, um die lokal hohe Konvergenzordnung einer Langschrittmethode zur L{\"o}sung von SDLCP's zu zeigen. Kapitel 6 f{\"u}hrt eine neue Methode zur L{\"o}sung von LCP's und SDLCP's mit Hilfe von Inneren-Punkte-Techniken ein. Dabei werden die Pfadfunktionen derart gew{\"a}hlt, dass alle Iterierten auf unzul{\"a}ssigen zentralen Pfaden liegen. Es wird globale und lokale Konvergenz des Verfahrens bewiesen.},
  subject      = {Innere-Punkte-Methode},
  language  = {de}
}
@phdthesis{BletzSiebert2002,
  author    = {Bletz-Siebert, Oliver},
  title     = {Homogeneous spaces with the cohomology of sphere products and compact quadrangles},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3994},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2002},
  abstract  = {We consider homogeneous spaces G/H with the same rational homotopy as a product of a 1-sphere and a (m+1)-sphere. We show that these spaces have also the rational cohomology of such a sphere product if H is connected and if the quotient has dimension m+2. Furthermore, we prove that if additionally the fundamental group of G/H is cyclic, then G/H is locally a product of a 1-torus and ofA/H, where A/H is a simply connected rational cohomology (m+1)-sphere (and hence classified). If H fails to be connected, then with U as the connected component of H the G-action on the covering space G/U of G/H has connected stabilizers, and the results apply to G/U. To show that under the assumptions above every natural number may be realized as the order of the group of connected components of H we calculate the cohomology of certain homogeneous spaces. We also determine the rational cohomology of the fibre bundle U-->G-->G/U if G/H meets the assumptions above. This is done by considering the respective Leray-Serre spectral sequence. The structure of the cohomology of U-->G-->G/U then gives a second proof for the structure of compact connected Lie groups acting transitively on spaces with the rational homotopy of a product of a 1-sphere and a (m+1)-sphere. Since a quotient of a homogeneous space with the same rational homotopy or cohomology as a product of a 1-sphere and a (m+1)-sphere is not simply connected, there often arises the question whether or not a considered fibre bundle or fibration is orientable. A large amount of space will therefore be given to the problem of showing that certain fibrations are orientable. For compact connected (m+2)-manifolds with cyclic fundamental groups and with the rational homotopy of a product of a 1-sphere and a (m+1)-sphere we show the following: if a connected Lie group acts transitively on the manifold, then the maximal compact subgroups are either transitive, or their orbits are simply connected rational cohomology spheres of codimension 1. Homogeneous spaces with the same rational cohomology or homotopy as a a product of a 1-sphere and a (m+1)-sphere play a role in the study of different types of geometrical objects. They appear for example as focal manifolds of isoparametric hypersurfaces with four distinct principal curvatures. Further examples of such spaces are the point spaces and the line spaces of compact connected generalized quadrangles. We determine the isometry groups of isoparametric hypersurfaces with 4 principal curvatures of multiplicities 1 and m which are transitive on the focal manifold with non-trivial fundamental group. Buildings were introduced by Jacques Tits to give interpretations of simple groups of Lie type. They are a far-reaching generalization of projective spaces, in particular a generalization of projective planes. There is another generalization of projective planes called generalized polygons. A projective plane is the same as a generalized triangle. The generalized polygons are also contained in the class of buildings: they are the buildings of rank 2. To compact quadrangles one can assign a pair of natural numbers called the topological parameters of the quadrangles. We treat the case k=1. It turns out that there are no other point-transitive compact connected Lie groups for (1,m)-quadrangles than the ones for the real orthogonal quadrangles. Furthermore, we solve the problem of three infinite series of group actions which Kramer left as open problems; there are no quadrangles with the homogeneous spaces in question as point spaces (up to maybe a finite number of small parameters in one of the three series).},
  subject      = {Homogener Raum},
  language  = {en}
}
@phdthesis{Grahl2002,
  author    = {Grahl, J{\"u}rgen},
  title     = {Blochsches Prinzip, L{\"u}ckenreihen und Semidualit{\"a}t},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3477},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2002},
  abstract  = {Ein bekanntes heuristisches Prinzip von A. Bloch beschreibt die Korrespondenz zwischen Kriterien f{\"u}r die Konstanz ganzer Funktionen und Normalit{\"a}tskriterien. In der vorliegenden Dissertation untersuchen wir die G{\"u}ltigkeit des Blochschen Prinzip bei L{\"u}ckenreihenproblemen sowie Zusammenh{\"a}nge zwischen Normalit{\"a}tsfragen und der Semidualit{\"a}t von einer bzw. von zwei Funktionen. Die ersten beiden Kapitel stellen die im folgenden ben{\"o}tigten Hilfsmittel aus der Nevanlinnaschen Wertverteilungstheorie und der Normalit{\"a}tstheorie bereit. Im dritten Kapitel beweisen wir ein neues Normalit{\"a}tskriterium f{\"u}r Familien holomorpher Funktionen, f{\"u}r die ein Differentialpolynom einer bestimmten Gestalt nullstellenfrei ist. Dies verallgemeinert fr{\"u}here Resultate von Hayman, Drasin, Langley und Chen \& Hua. Kapitel 4 ist dem Beweis eines unserer im folgenden wichtigsten Hilfsmittel gewidmet: eines tiefliegenden Konvergenzsatzes von H. Cartan {\"u}ber Familien von p-Tupeln holomorpher nullstellenfreier Funktionen, welche einer linearen Relation unterliegen. In Kapitel 5 werden die Konzepte der Dualit{\"a}t und Semidualit{\"a}t eingef{\"u}hrt und die Verbindung zu Normalit{\"a}tsfragen diskutiert. Die neuen Ergebnisse {\"u}ber L{\"u}ckenreihen finden sich im sechsten Kapitel. Der Schwerpunkt liegt hierbei zum einen auf sog. AP-L{\"u}ckenreihen, zum anderen auf allgemeinen Konstruktionsverfahren, mit denen sich neue semiduale L{\"u}ckenstrukturen aus bereits bekannten gewinnen lassen. Zahlreiche unserer Beweise beruhen wesentlich auf dem Satz von Cartan aus Kapitel 4. Im siebten Kapitel erweitern wir unsere Semidualit{\"a}tsuntersuchungen auf Mengen aus zwei Funktionen. Wir ziehen Normalit{\"a}tskriterien (vor allem das in Kapitel 3 bewiesene sowie den Satz von Cartan) heran, um spezielle Mengen als nichtsemidual zu identifizieren. Zuletzt konstruieren wir ein Beispiel einer semidualen Menge aus zwei Funktionen.},
  subject      = {Blochsches Prinzip},
  language  = {de}
}
@phdthesis{Wolfrom2002,
  author    = {Wolfrom, Martin},
  title     = {Isoparametric hypersurfaces with a homogeneous focal manifold},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-3505},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2002},
  abstract  = {The classification of isoparametric hypersurfaces in spheres with a homogeneous focal manifold is a project that has been started by Linus Kramer. It extends results by E. Cartan and Hsiang and Lawson. Kramer does most part of this classification in his Habilitationsschrift. In particular he obtains a classification for the cases where the homogeneous focal manifold is at least 2-connected. Results of E. Cartan, Dorfmeister and Neher, and Takagi also solve parts of the classification problem. This thesis completes the classification. We classify all closed isoparametric hypersurfaces in spheres with g>2 distinct principal curvatures one of whose multiplicities is 2 such that the lower dimensional focal manifold is homogeneous. The methods are essentially the same as in Kramer's 'Habilitationsschrift'. The cohomology of the focal manifolds in question is known. This leads to two topological classification problems, which are also solved in this thesis. We classify simply connected homogeneous spaces of compact Lie groups with the same integral cohomology ring as a product of spheres S^2 x S^m and m odd on the one hand and a truncated polynomial ring Q[a]/(a^m) with one generator of even degree and m > 1 as its rational cohomology ring on the other hand.},
  subject      = {Isoparametrische Hyperfl{\"a}che},
  language  = {en}
}
@phdthesis{Joachim2004,
  author    = {Joachim, Silvia},
  title     = {Regulatorketten in Butlergruppen},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-10438},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2004},
  abstract  = {Die fast vollst{\"a}ndig zerlegbaren Gruppen bilden eine Teilklasse der Butlergruppen. Das Konzept des Regulators, d.h. der Durchschnitt aller regulierenden Untergruppen, ist unverzichtbar f{\"u}r fast vollst{\"a}ndig zerlegbare Gruppen. Dieses Konzept l{\"a}sst sich in nat{\"u}rlicher Weise auf die ganze Klasse der Butlergruppen fortsetzen. Allerdings l{\"a}sst sich die Regulatorbildung im allgemeineren Fall der Butlergruppen a priori iterieren. Damit stellt sich erst einmal die Frage, ob es {\"u}berhaupt Butlergruppen gibt mit Regulatorketten, der L{\"a}nge gr{\"o}ßer als 1. Ein erstes Beispiel der L{\"a}nge 2 wurde 1997 von Lehrmann und Mutzbauer konstruiert. In dieser Dissertation wurden mit konzeptionell neuen Techniken Butlergruppen mit beliebiger vorgegebener endlicher Kettenl{\"a}nge angegeben. Grunds{\"a}tzliche Schwierigkeiten bei diesem Unterfangen resultieren aus dem Fehlen, bzw. der Unm{\"o}glichkeit, einer kanonischen Darstellung von Butlergruppen. Man verwendet die allseits gebrauchte Summendarstellung f{\"u}r Butlergruppen. Genau an dieser Stelle bedarf es v{\"o}llig neuer Methoden, verglichen mit den fast vollst{\"a}ndig zerlegbaren Gruppen mit ihrer kanonischen Regulatordarstellung. Alle Teilaufgaben bei der anstehenden Konstruktion von Butlergruppen, die f{\"u}r fast vollst{\"a}ndig zerlegbare Gruppen Standard sind, werden hierbei problematisch, u.a. die Bildung reiner H{\"u}llen, die Bestimmung regulierender Untergruppen und die Regulatorbildung.},
  subject      = {Butlergruppe},
  language  = {de}
}
@phdthesis{Kurniawan2009,
  author    = {Kurniawan, Indra},
  title     = {Controllability Aspects of the Lindblad-Kossakowski Master Equation : A Lie-Theoretical Approach},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-48815},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2009},
  abstract  = {One main task, which is considerably important in many applications in quantum control, is to explore the possibilities of steering a quantum system from an initial state to a target state. This thesis focuses on fundamental control-theoretical issues of quantum dynamics described by the Lindblad-Kossakowski master equation which arises as a bilinear control system on some underlying real vector spaces, e.g controllability aspects and the structure of reachable sets. Based on Lie-algebraic methods from nonlinear control theory, the thesis presents a unified approach to control problems of finite dimensional closed and open quantum systems. In particular, a simplified treatment for controllability of closed quantum systems as well as new accessibility results for open quantum systems are obtained. The main tools to derive the results are the well-known classifications of all matrix Lie groups which act transitively on Grassmann manifolds, and respectively, on real vector spaces without the origin. It is also shown in this thesis that accessibiity of the Lindblad-Kossakowski master equation is a generic property. Moreover, based on the theoretical accessibility results, an algorithm is developed to decide when the Lindblad-Kossakowski master equation is accessible.},
  subject      = {Kontrolltheorie},
  language  = {en}
}
@phdthesis{Koenig2014,
  author    = {K{\"o}nig, Joachim},
  title     = {The inverse Galois problem and explicit computation of families of covers of \(\mathbb{P}^1\mathbb{C}\) with prescribed ramification},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-100143},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2014},
  abstract  = {In attempting to solve the regular inverse Galois problem for arbitrary subfields K of C (particularly for K=Q), a very important result by Fried and V{\"o}lklein reduces the existence of regular Galois extensions F|K(t) with Galois group G to the existence of K-rational points on components of certain moduli spaces for families of covers of the projective line, known as Hurwitz spaces. In some cases, the existence of rational points on Hurwitz spaces has been proven by theoretical criteria. In general, however, the question whether a given Hurwitz space has any rational point remains a very difficult problem. In concrete cases, it may be tackled by an explicit computation of a Hurwitz space and the corresponding family of covers. The aim of this work is to collect and expand on the various techniques that may be used to solve such computational problems and apply them to tackle several families of Galois theoretic interest. In particular, in Chapter 5, we compute explicit curve equations for Hurwitz spaces for certain families of \(M_{24}\) and \(M_{23}\). These are (to my knowledge) the first examples of explicitly computed Hurwitz spaces of such high genus. They might be used to realize \(M_{23}\) as a regular Galois group over Q if one manages to find suitable points on them. Apart from the calculation of explicit algebraic equations, we produce complex approximations for polynomials with genus zero ramification of several different ramification types in \(M_{24}\) and \(M_{23}\). These may be used as starting points for similar computations. The main motivation for these computations is the fact that \(M_{23}\) is currently the only remaining sporadic group that is not known to occur as a Galois group over Q. We also compute the first explicit polynomials with Galois groups \(G=P\Gamma L_3(4), PGL_3(4), PSL_3(4)\) and \(PSL_5(2)\) over Q(t). Special attention will be given to reality questions. As an application we compute the first examples of totally real polynomials with Galois groups \(PGL_2(11)\) and \(PSL_3(3)\) over Q. As a suggestion for further research, we describe an explicit algorithmic version of "Algebraic Patching", following the theory described e.g. by M. Jarden. This could be used to conquer some problems regarding families of covers of genus g>0. Finally, we present explicit Magma implementations for several of the most important algorithms involved in our computations.},
  subject      = {Galoistheorie},
  language  = {en}
}