A dictionary of modular threefolds

The thesis deals with the modularity conjecture for three-dimensional Calabi-Yau varieties. This is a generalization of the work of A. Wiles and others on modularity of elliptic curves. Modularity connects the number of points on varieties with coefficients of certain modular forms. In chapter 1 we collect the basics on arithmetic on Calabi-Yau manifolds, including general modularity results and strategies for modularity proofs. In chapters 2, 3, 4 and 5 we investigate examples of modular Calabi-Yau threefolds, including all examples occurring in the literature and many new ones. Double octics, i.e. Double coverings of projective 3-space branched along an octic surface, are studied in detail. In chapter 6 we deal with examples connected with the same modular forms. According to the Tate conjecture there should be correspondences between them. Many correspondences are constructed explicitly. We finish by formulating conjectures on the occurring newforms, especially their levels. In the appendices we compile tables of coefficients of weight 2 and weight 4 newforms and many examples of double octics.

The swelling behaviour of polyelectrolyte networks

Being basic ingredients of numerous daily-life products with significant industrial importance as well as basic building blocks for biomaterials, charged hydrogels continue to pose a series of unanswered challenges for scientists even after decades of practical applications and intensive research efforts. Despite a rather simple internal structure it is mainly the unique combination of short- and long-range forces which render scientific investigations of their characteristic properties to be quite difficult. Hence early on computer simulations were used to link analytical theory and empirical experiments, bridging the gap between the simplifying assumptions of the models and the complexity of real world measurements. Due to the immense numerical effort, even for high performance supercomputers, system sizes and time scales were rather restricted until recently, whereas it only now has become possible to also simulate a network of charged macromolecules. This is the topic of the presented thesis which investigates one of the fundamental and at the same time highly fascinating phenomenon of polymer research: The swelling behaviour of polyelectrolyte networks. For this an extensible simulation package for the research on soft matter systems, ESPResSo for short, was created which puts a particular emphasis on mesoscopic bead-spring-models of complex systems. Highly efficient algorithms and a consistent parallelization reduced the necessary computation time for solving equations of motion even in case of long-ranged electrostatics and large number of particles, allowing to tackle even expensive calculations and applications. Nevertheless, the program has a modular and simple structure, enabling a continuous process of adding new potentials, interactions, degrees of freedom, ensembles, and integrators, while staying easily accessible for newcomers due to a Tcl-script steering level controlling the C-implemented simulation core. Numerous analysis routines provide means to investigate system properties and observables on-the-fly. Even though analytical theories agreed on the modeling of networks in the past years, our numerical MD-simulations show that even in case of simple model systems fundamental theoretical assumptions no longer apply except for a small parameter regime, prohibiting correct predictions of observables. Applying a "microscopic" analysis of the isolated contributions of individual system components, one of the particular strengths of computer simulations, it was then possible to describe the behaviour of charged polymer networks at swelling equilibrium in good solvent and close to the Theta-point by introducing appropriate model modifications. This became possible by enhancing known simple scaling arguments with components deemed crucial in our detailed study, through which a generalized model could be constructed. Herewith an agreement of the final system volume of swollen polyelectrolyte gels with results of computer simulations could be shown successfully over the entire investigated range of parameters, for different network sizes, charge fractions, and interaction strengths. In addition, the "cell under tension" was presented as a self-regulating approach for predicting the amount of swelling based on the used system parameters only. Without the need for measured observables as input, minimizing the free energy alone already allows to determine the the equilibrium behaviour. In poor solvent the shape of the network chains changes considerably, as now their hydrophobicity counteracts the repulsion of like-wise charged monomers and pursues collapsing the polyelectrolytes. Depending on the chosen parameters a fragile balance emerges, giving rise to fascinating geometrical structures such as the so-called pear-necklaces. This behaviour, known from single chain polyelectrolytes under similar environmental conditions and also theoretically predicted, could be detected for the first time for networks as well. An analysis of the total structure factors confirmed first evidences for the existence of such structures found in experimental results.

New design of polyphenylene dendrimers for full-color light-emitting diodes

Dendritic systems, and in particular polyphenylene dendrimers, have recently attracted considerable attention from the synthetic organic chemistry community, as well as from photophysicists, particularly in view of the search for synthetic model analogies to photoelectric materials to fabricate organic light-emitting diodes (OLEDs), and even more advanced areas of research such as light-harvesting system, energy transfer and non-host device. Geometrically, dendrimers are unique systems that consist of a core, one or more dendrons, and surface groups. The different parts of the macromolecule can be selected to give the desired optoelectronic and processing properties. Compared to small molecular or polymeric light-emitting materials, these dendritic materials can combine the benefits of both previous classes. The high molecular weights of these dendritic macromolecules, as well as the surface groups often attached to the distal ends of the dendrons, can improve the solution processability, and thus can be deposited from solution by simple processes such as spin-coating and ink-jet printing. Moreover, even better than the traditional polymeric light-emitting materials, the well-defined monodisperse distributed dendrimers possess a high purity comparable to that of small molecules, and as such can be fabricated into high performance OLEDs. Most importantly, the emissive chromophores can be located at the core of the dendrimer, within the dendrons, and/or at the surface of the dendrimers because of their unique dendritic architectures. The different parts of the macromolecule can be selected to give the desired optoelectronic and processing properties. Therefore, the main goals of this thesis are the design and synthesis, characterization of novel functional dendrimers, e.g. polytriphenylene dendrimers for blue fluorescent, as well as iridium(III) complex cored polyphenylene dendrimers for green and red phosphorescent light emitting diodes. In additional to the above mentioned advantages of dendrimer based OLEDs, the modular molecular architecture and various functionalized units at different locations in polyphenylene dendrimers open up a tremendous scope for tuning a wide range of properties in addition to color, such as intermolecular interactions, charge mobility, quantum yield, and exciton diffusion. In conclusion, research into dendrimer containing OLEDs combines fundamental aspects of organic semiconductor physics, novel and highly sophisticated organic synthetic chemistry and elaborate device technology.rn

DNA-Polymer-Hybride als molekulare Bausteine

Die DNA hat sich durch die herausstechende Eigenschaft zur Selbstorganisation in den Naturwissenschaften zu einem beliebten Werkzeug entwickelt. In dieser Arbeit wurde die Oligonukleotidselbsterkennung zum Aufbau komplexer Multiblockcopolymere genutzt. Dabei dienten komplementäre einzelsträngige Oligonukleotidsequenzen (ssDNA) als adressierbare Verbindungsstücke zwischen synthetischen Blöcken. Als Bausteine wurden asymmetrische Dreiblockcopolymere der Form DNA1-Polymer-DNA2 aus einer flexiblen Polymereinheit (PEO bzw. PPO) die an beiden Enden mit unterschiedlichen Oligonukleotidsequenzen „funktionalisiert“ ist, verwendet. Diese Bausteine konnten durch die Kombination von Festphasensynthese der Oligonukleotide und Blockkopplung dargestellt werden. Die Oligonukleotidsequenzen wurden so gewählt, dass deren Hybridisierung zu einer bei Raumtemperatur stabilen Verbindung führt. Durch die Verwendung dieser Bausteine erhält man ein modulares System, dass sich durch seine hohe Flexibilität auszeichnet. Aus den dargestellten Dreiblockcopolymeren konnten verschiedene alternierende Multiblockcopolymere aufgebaut werden, wobei die Anzahl der Blöcke (von 11 bis 15) und das PEO / PPO- Verhältnis variiert wurden. Derartige Strukturen sind auf der Grundlage chemischer Synthesen unerreichbar. Die Flexibilität dieses modularen Systems konnte gezeigt werden, indem einzelne Blockbausteine zur Strukturaufklärung einfach ausgetauscht oder weggelassen werden konnten. Durch geeignete Wahl der DNA-Sequenzen konnte zusätzlich das Polymerisationsverhalten dieser Bauelemente untersucht werden. Die Integration längerer kettensteifer DNA-Abschnitte in die Multiblockstrukturen erfolgte durch die Verwendung teilkomplementärer Oligonukleotide. Diese bieten den Vorteil, dass bis zu einer Größe von etwa 150 bp sowohl die Länge als auch die Sequenz der Doppelstrangabschnitte und sticky-ends frei variiert werden können. Die biosynthetischen Dreiblockcopolymere dienten hier als Linkermoleküle zwischen den einzelnen dsDNA-Blöcken. Nach diesem Konzept wurde ein Nonamer als Modellsystem eines mehrfach gebrochenen Stäbchens synthetisiert. Außerdem wurden mit Hilfe der Polymerase Kettenreaktion (PCR) semiflexible DNA Abschnitte erzeugt. Durch die Wahl des Synthesewegs konnte sowohl die Länge der semiflexiblen Einheit als auch die Länge und die Sequenz des sticky-ends variiert werden. Anhand dieser Modellverbindungen wurde dann das Hybridisierungsverhalten in Abhängigkeit der Linker- und Segmentlängen untersucht.

New synthetic strategies towards indolizines and pyrroles

Indolizines and pyrroles are considered as “privileged” structures since their skeletons were found in many biologically active natural products and they possess a wide range of pharmaceutical properties. Syntheses of these small drug-like molecules are very important in medicinal chemistry. However, most existent methodologies are usually limited to specific substitution patterns or require impractical starting materials or expensive catalysts. Therefore, developing new methodologies for the synthesis of indolizines and pyrroles from commercially available or readily accessible sources is highly desirable.rnIn this PhD thesis, several methods has been described for the synthesis of indolizines and pyrroles. In the first part, indolizines carrying substituents in positions 1-3 were synthesized via a formal [3+2]-cycloaddition of pyridinium ylides and nitroalkenes. Pyridinium salts were prepared by N-alkylation of pyridines with cyanohydrin triflates which could be prepared from corresponding aldehydes via a Strecker reaction followed by O-triflylation. Nitroalkenes were simply prepared from the corresponding aldehydes and nitroalkanes in a nitroaldol condensation. Overall, this modular approach allows to construct the indolizine framework with various substitution patterns starting from a pyridine, two different aldehydes and a nitroalkane. In contrast to reported methods, the produced indolizines do not have to contain an electron-withdrawing group.rnIt has also been found that nitrile-stabilized 2-alkylpyridinium ylides cyclize to unstable 2-aminoindolizines via an intramolecular 5-exo-dig cyclization. Using an in situ acetylation of the amino group, N-protected 2-aminoindolizines could be synthesized. As a less common substitution pattern, indolizines carrying substituents in positions 5–8 were synthesized from enones and 2-(1H-pyrrol-1-yl)nitriles obtained from α-aminonitriles using a modified Paal-Knorr pyrrole synthesis. The decoration of the pyridine unit in the indolizine skeleton has been achieved by a one-pot conjugate addition/cycloaromatization sequence.rnIn the second part of the thesis, the diversity-oriented synthesis of pyrroles from 3,5-diaryl substituted 2H-pyrrole-2-carbonitriles (cyanopyrrolines) obtained in a cyclocondensation of enones with aminoacetonitrile hydrochloride is being discussed. 2,4-Di-, 2,3,5-trisubstituted pyrroles, pyrrole-2-carbonitriles and 2,2’-bipyrroles were synthesized in a one- or two-step protocol. While the microwave-assisted thermal elimination of HCN from cyanopyrrolines gave 2,4-disubstituted pyrroles, DDQ-oxidation of the same intermediates furnished pyrrole-2-carbonitriles. Furthermore, 2,3,5-trisubstituted pyrroles were obtained via a C-2-alkylation of the deprotonated cyanopyrrolines followed by the elimination of HCN. Finally, it has also been found that tetraaryl substituted 2,2’-bipyrroles could be synthesized by the oxidative dimerization of cyanopyrrolines using copper (II) acetate at 100 °C.rn