Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

On Random Planar Curves and Their Scaling Limits

Planar curves arise naturally as interfaces between two regions of the plane. An important part of statistical physics is the study of lattice models. This thesis is about the interfaces of 2D lattice models. The scaling limit is an infinite system limit which is taken by letting the lattice mesh decrease to zero. At criticality, the scaling limit of an interface is one of the SLE curves (Schramm-Loewner evolution), introduced by Oded Schramm. This family of random curves is parametrized by a real variable, which determines the universality class of the model. The first and the second paper of this thesis study properties of SLEs. They contain two different methods to study the whole SLE curve, which is, in fact, the most interesting object from the statistical physics point of view. These methods are applied to study two symmetries of SLE: reversibility and duality. The first paper uses an algebraic method and a representation of the Virasoro algebra to find common martingales to different processes, and that way, to confirm the symmetries for polynomial expected values of natural SLE data. In the second paper, a recursion is obtained for the same kind of expected values. The recursion is based on stationarity of the law of the whole SLE curve under a SLE induced flow. The third paper deals with one of the most central questions of the field and provides a framework of estimates for describing 2D scaling limits by SLE curves. In particular, it is shown that a weak estimate on the probability of an annulus crossing implies that a random curve arising from a statistical physics model will have scaling limits and those will be well-described by Loewner evolutions with random driving forces.

Paradise Reconsidered : A Study of the Ophite Myth and Ritual and their Relationship to Sethianism

This thesis examines the mythology in and social reality behind a group of texts from the Nag Hammadi and related literature, to which certain leaders of the early church attached the label, Ophite, i.e., snake people. In the mythology, which essentially draws upon and rewrites the Genesis paradise story, the snake's advice to eat from the tree of knowledge is positive, the creator and his angels are demonic beasts and the true godhead is depicted as an androgynous heavenly projection of Adam and Eve. It will be argued that this unique mythology is attested in certain Coptic texts from the Nag Hammadi and Berlin 8502 Codices (On the Origin of the World, Hypostasis of the Archons, Apocryphon of John, Eugnostos, Sophia of Jesus Christ), as well as in reports by Irenaeus (Adversus Haereses 1.30), Origen (Contra Celsum 6.24-38) and Epiphanius (Panarion 26). It will also be argued that this so-called Ophite evidence is essential for a proper understanding of Sethian Gnosticism, often today considered one of the earliest forms of Gnosticism; there seems to have occurred a Sethianization of Ophite mythology. I propose that we replace the current Sethian Gnostic category by a new one that not only adds texts that draw upon the Ophite mythology alongside these Sethian texts, but also arranges the material in smaller typological units. I also propose we rename this remodelled and expanded Sethian corpus "Classic Gnostic." I have divided the thesis into four parts: (I) Introduction; (II) Myth and Innovation; (III) Ritual; and (IV) Conclusion. In Part I, the sources and previous research on Ophites and Sethians will be examined, and the new Classic Gnostic category will be introduced to provide a framework for the study of the Ophite evidence. Chapters in Part II explore key themes in the mythology of our texts, first by text comparison (to show that certain texts represent the Ophite mythology and that this mythology is different from Sethianism), and then by attempting to unveil social circumstances that may have given rise to such myths. Part III assesses heresiological claims of Ophite rituals, and Part IV is the conclusion.

Phylogenetics of Myrmica ants and their social parasites

Understanding the overwhelming diversity of life calls for complex organisational schemes. The field of systematics may thus be seen as the cornerstone of evolutionary biology. In the last few decades, systematics has been rejuvenated through the introduction of molecular methods such as DNA barcoding and multi-gene phylogenetic approaches. These methods may shed new light on established taxonomic ideas and problems. For example, the classification of ants has aroused much debate due to reinterpretation of morphological characters or contradictions between molecular data and morphology. Only in the last few years a consensus was reached regarding the phylogeny of ant subfamilies. However, the situation remains deplorable for lower taxonomic ranks such as subfamilies, tribes and genera. This thesis describes the systematics and evolution of the Holarctic ant genus Myrmica and the tribe to which it belongs, Myrmicini. Using barcoding, molecular-phylogenetic data and divergence time estimations, it addresses questions regarding the taxonomy, morphology and biogeography of this group. Furthermore, the interrelationships between socially parasitic Myrmica species and their hosts (other species in the genus) were inferred. The phylogeny suggests that social parasitism evolved several times in Myrmica. Finally, this thesis investigated whether coevolution shaped the phylogeny of socially parasitic Maculinea butterflies that live inside Myrmica colonies. No evidence was found for coevolution.

Neurotrophic factors and their receptors

Four GDNF ligands (GDNF, neurturin, artemin and persephin), and mesencephalic astrocyte-derived neurotrophic factor (MANF) and conserved dopamine neurotrophic factor (CDNF) protect midbrain dopaminergic neurons that degenerate in Parkinson's disease. Each GDNF ligand binds a specific coreceptor GDNF family receptor α (GFRα), leading to the formation of a heterotetramer complex, which then interacts with receptor tyrosine kinase RET, the signalling receptor. The present thesis describes the structural and biochemical characterization of the GDNF2-GFRα12 complex and the MANF and CDNF proteins. Previous and current mutation data and comparison between GDNF-GFRα1 and artemin-GFRα3 binding interfaces show that N162GFRα1, I175GFRα1, V230GFRα1, Y120GDNF and L114GDNF are the specificity determinants among different ligand-coreceptor pairs. The structure suggests that sucrose octasulphate, a heparin mimic, interacts with a region R190-K202 within domain 2 of GFRα1. Mutating these residues on the GFRα1 surface, which are not in the GDNF binding region, affected RET phosphorylation, which provides a putative RET binding region in domain 2 and 3 of GFRα1. The structural comparison of the GDNF-GFRα1 and artemin-GFRα3 complexes shows a difference in bend angle between the ligand monomers. This variation in bend angle of the ligand may affect the kinetics of RET phosphorylation. To confirm that the difference is not due to crystallization artefacts, I crystallized the GDNF-GFRα1 complex without SOS in different cell dimensions. The structure of the second GDNF-GFRα1 complex is very similar to the previous one, suggesting that the difference between the artemin-GFRα3 and GDNF-GFRα1 complexes are intrinsic, not due to crystal packing. Finally, MANF and CDNF are bifunctional proteins with extracellular neurotrophic activity and ER resident cytoprotective role. The crystal structures of MANF and CDNF are presented here. Intriguingly, the structures of both the neurotrophic factors do not show structural similarity to any of previously known growth factor superfamilies; instead they are similar to saposins, the lipid-binding proteins. The N-terminal domain of MANF and CDNF contain conserved lysines and arginines on its surface, which may interact with negatively charged head groups of phospholipids, as saposins do. Thus MANF and CDNF may provide neurotrophic activities by interacting with a lipo-receptor. The structure of MANF shows a CXXC motif forming internal disulphide bridge in the natively unfolded C-terminus. This motif is common to reductases and disulphide isomerases. It is thus tempting to speculate that the CXXC motif of MANF and CDNF may be involved in oxidative protein folding, which may explain its cytoprotective role in the ER.