## Fachbereich Mathematik

- Vector bundles on elliptic curves and factors of automorphy (2005)
- The Atiyah’s results on classification of vector bundles on elliptic curves are translated to the language of factors of automorphy.

- Intersection Theory of the Tropical Moduli Spaces of Curves (2007)
- Tropical geometry is a very new mathematical domain. The appearance of tropical geometry was motivated by its deep relations to other mathematical branches. These include algebraic geometry, symplectic geometry, complex analysis, combinatorics and mathematical biology. In this work we see some more relations between algebraic geometry and tropical geometry. Our aim is to prove a one-to-one correspondence between the divisor classes on the moduli space of n-pointed rational stable curves and the divisors of the moduli space of n-pointed abstract tropical curves. Thus we state some results of the algebraic case first. In algebraic geometry these moduli spaces are well understood. In particular, the group of divisor classes is calculated by S. Keel. We recall the needed results in chapter one. For the proof of the correspondence we use some results of toric geometry. Further we want to show an equality of the Chow groups of a special toric variety and the algebraic moduli space. Thus we state some results of the toric geometry as well. This thesis tries to discover some connection between algebraic and tropical geometry. Thus we also need the corresponding tropical objects to the algebraic objects. Therefore we give some necessary definitions such as fan, tropical fan, morphisms between tropical fans, divisors or the topical moduli space of all n-marked tropical curves. Since we need it, we show that the tropical moduli space can be embedded as a tropical fan. After this preparatory work we prove that the group of divisor classes in v classical algebraic geometry has it equivalence in tropical geometry. For this it is useful to give a map from the group of divisor classes of the algebraic moduli space to the group of divisors of the tropical moduli space. Our aim is to prove the bijectivity of this map in chapter three. On the way we discover a deep connection between the algebraic moduli space and the toric variety given by the tropical fan of the tropical moduli space.

- Asymptotics for a thin elastic fiber in contact with a rigid body (2010)
- In this work a 3-dimensional contact elasticity problem for a thin fiber and a rigid foundation is studied. We describe the contact condition by a linear Robin-condition (by meaning of the penalized and linearized non-penetration and friction conditions). The dimension of the problem is reduced by an asymptotic approach. Scaling the Robin parameters appropriately we obtain a recurrent chain of Neumann type boundary value problems which are considered only in the microscopic scale. The problem for the leading term is a homogeneous Neumann problem, hence the leading term depends only on the slow variable. This motivates the choice of a multiplicative ansatz in the asymptotic expansion. The theoretical results are illustrated with numerical examples performed with a commercial finite-element software-tool.

- Computation of the central elements and centralizers of sets of elements in non-commutative polynomial algebras (2006)
- In this thesis we present the implementation of libraries center.lib and perron.lib for the non-commutative extension Plural of the Computer Algebra System Singular. The library center.lib was designed for the computation of elements of the centralizer of a set of elements and the center of a non-commutative polynomial algebra. It also provides solutions to related problems. The library perron.lib contains a procedure for the computation of relations between a set of pairwise commuting polynomials. The thesis comprises the theory behind the libraries, aspects of the implementation and some applications of the developed algorithms. Moreover, we provide extensive benchmarks for the computation of elements of the center. Some of our examples were never computed before.

- Four Generations of Asset Pricing Models and Volatility Dynamics (2007)
- The scope of this diploma thesis is to examine the four generations of asset pricing models and the corresponding volatility dynamics which have been devepoled so far. We proceed as follows: In chapter 1 we give a short repetition of the Black-Scholes first generation model which assumes a constant volatility and we show that volatility should not be modeled as constant by examining statistical data and introducing the notion of implied volatility. In chapter 2, we examine the simplest models that are able to produce smiles or skews - local volatility models. These are called second generation models. Local volatility models model the volatility as a function of the stock price and time. We start with the work of Dupire, show how local volatility models can be calibrated and end with a detailed discussion of the constant elasticity of volatility model. Chapter 3 focuses on the Heston model which represents the class of the stochastic volatility models, which assume that the volatility itself is driven by a stochastic process. These are called third generation models. We introduce the model structure, derive a partial differential pricing equation, give a closed-form solution for European calls by solving this equation and explain how the model is calibrated. The last part of chapter 3 then deals with the limits and the mis-specifications of the Heston model, in particular for recent exotic options like reverse cliquets, Accumulators or Napoleons. In chapter 4 we then introduce the Bergomi forward variance model which is called fourth generation model as a consequence of the limits of the Heston model explained in chapter 3. The Bergomi model is a stochastic local volatility model - the spot price is modeled as a constant elasticity of volatility diffusion and its volatility parameters are functions of the so called forward variances which are specified as stochastic processes. We start with the model specification, derive a partial differential pricing equation, show how the model has to be calibrated and end with pricing examples and a concluding discussion.

- A Survey of Earliest Arrival Flows and a Study of the Series-Parallel Case (2009)
- This work is concerned with dynamic flow problems, especially maximal dynamic flows and earliest arrival flows - also called universally maximal flows. First of all, a survey of known results about existence, computation and approximation of earliest arrival flows is given. For the special case of series-parallel graphs a polynomial algorithm for computing maximal dynamic flows is presented and this maximal dynamic flow is proven to be an earliest arrival flow.

- Weight-Constrained Minimum Spanning Tree Problem (2007)
- In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.

- Polyhedral Analysis of Hub Center Problems (2003)
- A hub location problem consists of locating p hubs in a network in order to collect and consolidate flow between node pairs. This thesis deals with the uncapacitated single allocation p-hub center problem (USApHCP) as a special type of hub location problem with min max objective function. Using the so-called radius formulation of the problem, the dimension of the polyhedron of USApHCP is derived. The formulation constraints are investigated to find out which of these define facets. Then, three new classes of facet-defining inequalities are derived. Finally, efficient procedures to separate facets in a branch-and-cut algorithm are proposed. The polyhedral analysis of USApHCP is based on a tight relation to the uncapacitated facility location problem (UFL). Hence, many results stated in this thesis also hold for UFL.

- Wavelet-based Adaptive Multiresolution Tools Applied to Speech Recognition (2006)
- * naive examples which show drawbacks of discrete wavelet transform and windowed Fourier transform; * adaptive partition (with a 'best basis' approach) of speech-like signals by means of local trigonometric bases with orthonormal windows. * extraction of formant-like features from the cosine transform; * further proceedingings for classification of vowels or voiced speech are suggested at the end.

- Das Verhalten ausgewählter technischer Indikatoren bei simulierten Finanzzeitreihen (2002)
- In der Arbeit sollen ausgewählte technische Indikatoren und deren Handelsstrategien hinsichtlich ihres Verhaltens bzw. ihrer Profitabilität in verschiedenen Marktphasen untersucht werden. Um das Argument, dass die Indikatoren selbst erfüllend sind, zu entkräften, werden Finanzzeitreihen simuliert und die Indikatoren auf diese angewendet und ausgewertet. Zu diesem Zweck wird zu gegebenen echten Kursdaten ein finanzzeitreihenanalytisches Modell angepasst. Dieses wird zur Simulation von Finanzzeitreihen und damit zur Auswertung der Indikatoren verwendet werden. Durch geeignete Auswahlverfahren sollen verschiedene Handelsstrategien zu Strategien kombiniert werden, um ein besseres Ertrag/Risikoverhältnis zu erreichen als dies bei einzelnen Strategien der Fall wäre.