On the geometry of p-origamis and beyond

The main topic of this thesis is the study of a special class of translation surfaces called normal origamis. The theory of translation surfaces is an active research area with applications in various fields such as dynamical systems, algebraic geometry, and geometric group theory. Normal origamis are surfaces with a maximal symmetry group and induce normal covers of the torus T. We focus on p-origamis, where the deck transformation groups of the torus covers are p-groups, and answer the questions: Which strata contain p-origamis? Does already the deck transformation group determine the stratum? We then turn toward the study of Veech groups of certain normal origamis. These groups are the stabilizer groups of an origami under an SL(2,Z)-action. We are especially interested in the question, whether the occurring Veech groups are congruence groups. The SL(2,Z)-orbits on normal origamis are closely related to the group-theoretic concept of T_2-systems. We investigate this relationship and transfer group-theoretic results to the geometric setting. Cylinder decompositions are an important concept occurring in different contexts within this thesis. Geminal origamis exhibit special cylinder decompositions. Apisa and Wright asked whether geminal origamis are cyclic covers of the surface (2 x 2)-torus. We use methods from group theory to answer this question partially. This thesis contains results of the author's research articles [FT20] and [The21].Im Zentrum dieser Dissertation steht das Studium normaler Origamis, einer Familie von Translationsflächen. Seit 40 Jahren sind Translationsflächen Gegenstand aktiver mathematischer Forschung mit Anwendungen in diversen mathematischen Bereichen wie algebraischer Geometrie und geometrischer Gruppentheorie. Normale Origamis haben eine maximale Symmetriegruppe und definieren normale Überlagerungen des Torus. Zunächst untersuchen wir p-Origamis, d.h. normale Origamis mit einer p-Gruppe als Decktransformationsgruppe. Wir beantworten die Fragen, welche Strata p-Origamis enthalten und ob die Decktransformationsgruppe bereits das Stratum festlegt. Des Weiteren betrachten wir Veechgruppen bestimmter normaler Origamis. Diese Gruppen sind Stabilisatoren eines Origamis unter einer SL(2,Z)-Wirkung. Unter anderem untersuchen wir die Fragen, ob und wann die Veechgruppen normaler Origamis Kongruenzgruppen sind. Zudem diskutieren wir den Zusammenhang zwischen den SL(2,Z)-Bahnen normaler Origamis und dem gruppentheoretischen Konzept der T_2-Systeme. Zylinderzerlegungen sind ein wichtiges Konzept in der Theorie der Translationsflächen, welches wir an verschiedenen Stellen in dieser Arbeit verwenden. Geminale Origamis sind Origamis mit sehr speziellen Zylinderzerlegungen. Unter gewissen Voraussetzungen beantworten wir die Frage, ob geminale Origamis den (2 x 2)-Torus zyklisch überlagern. Diese Dissertation enthält Ergebnisse aus folgenden Publikationen der Autorin [FT20] und [The21].German Research Foundation (DFG): SFB-TRR 195 Symbolic Tools in Mathematics and their Application

Rational approximations, multidimensional continued fractions and lattice reduction

We first survey the current state of the art concerning the dynamical properties of multidimensional continued fraction algorithms defined dynamically as piecewise fractional maps and compare them with algorithms based on lattice reduction. We discuss their convergence properties and the quality of the rational approximation, and stress the interest for these algorithms to be obtained by iterating dynamical systems. We then focus on an algorithm based on the classical Jacobi--Perron algorithm involving the nearest integer part. We describe its Markov properties and we suggest a possible procedure for proving the existence of a finite ergodic invariant measure absolutely continuous with respect to Lebesgue measure.Comment: 30 pages, 4 figure

Do dried blood spots have the potential to support result management processes in routine sports drug testing?—Part 3: LC–MS/MS‐based peptide analysis for dried blood spot sampling time point estimation

Along with the recent acknowledgement of the World Anti-Doping Agency to use dried blood spot (DBS) samples for routine doping control purposes, there have been propositions to use DBS as a matrix that allows regular proactive remotely supervised self-sampling, providing potential longitudinal monitoring of an athlete's exposure to doping agents. However, several organizational aspects have to be considered before implementation, such as the verification of the sample collections time point. Based on a previous untargeted proteomics workflow utilizing liquid chromatography–high-resolution mass spectrometry (LC–HRMS) to identify protein/peptide markers to define the time since deposition of a bloodstain, the aim of the current study was to develop a targeted LC–HRMS/MS analytical method for promising peptidic target analytes. A long-term DBS storage experiment was carried out over a 3-month period (sample collection time points: 0, 2, 4, 7, 14, 21, 28, 42, 56, 70, 84 and 91 days) with DBS samples of 10 volunteers for longitudinal investigation of signal abundance changes of targeted peptide sequences at different storage temperatures (room temperature [RT], 4°C and −20°C). Prior to experimental analysis, LC–HRMS/MS method characteristics were successfully assessed, including intraday precision, carryover and sample extract stability. For estimation of DBS sample collection time points, ratios of two peptides that originate from the same protein prior to tryptic digestion were created. Two targeted peptide area ratios were found to significantly increase after being stored at RT for 28 days, representing potential markers for future use in routine doping controls that contribute to advancing complementary avenues in anti-doping