On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Titanium Complexes for the Polymerisation of Ethene and Caprolactone

The literature part of the thesis mainly reviews the results of the use of titanium catalysts for ethene and caprolactone polymerisation. The behaviour of titanium catalysts bearing phenoxy-imino ligands has been the focus of more detailed investigations in ethene polymerisation. Reasons for the production of multimodal polyethene for a range of catalysts are also given. The experimental part of the thesis is divided into two sections based on the monomers used in the polymerisations: Part A (ethene) and part B (caprolactone). Part A: Titanium(IV) complexes bearing phenoxy-imino ligands are known to possess high ethene polymerisation activities after MAO activation. Depending on the ligand, the activities of the catalysts in polymerisation can vary between 1 and 44000 kgPE/(mol*cat*h*bar). Depending on the polymerisation temperature and the electronic and steric properties of the catalyst ligands, low to high molar mass values and uni- and multimodal polydispersity values can been observed. In order to discover the reasons for these differences, 22 titanium(IV) complexes containing differently substituted phenoxy-imino derivatives as di- and tetradentate ligands were synthesised with high yields and used as homogeneous catalysts in ethene polymerisations. Computational methods were used to predict the geometry of the synthesised complexes and their configuration after activation. Based on the results obtained, the geometry of the catalyst together with the ligand substituents seem to play a major role in defining the catalytic activity. Novel titanium(IV) complexes bearing malonate ligands were also synthesised. Malonates are considered to be suitable ligand pre-cursors since they can be produced by the simple reaction of any primary or secondary alcohol with malonylchloride, and thus they are easily modifiable. After treatment with MAO these complexes had polymerisation activities between 10 and 50 kgPE/(mol*cat*h*bar) and surprisingly low polydispersity values when compared with similar types of catalysts bearing the O?O chelate ligand. Part B: One of the synthesis routes in the preparation of the above mentioned phenoxy-imino titanium dichloride complexes involved the use of Ti(NMe2)4 with a range of salicylaldimine type compounds. On reaction, these two compounds formed an intermediate product selectively and quantitatively which was active in the ring-opening polymerisation of caprolactone. Several mono-anionic alcoholates were also combined with Ti(NMe2)4 in different molar ratios and used as catalysts. Full conversion of the monomer was achieved within 15 minutes with catalysts having a co-ordination number of 4 while after 22 hours full conversion was achieved with catalysts having a co-ordination number of 6.

Parameters Affecting the Extraction of Polycyclic Aromatic Hydrocarbons with Pressurised Hot Water

Pressurised hot water extraction (PHWE) exploits the unique temperature-dependent solvent properties of water minimising the use of harmful organic solvents. Water is environmentally friendly, cheap and easily available extraction medium. The effects of temperature, pressure and extraction time in PHWE have often been studied, but here the emphasis was on other parameters important for the extraction, most notably the dimensions of the extraction vessel and the stability and solubility of the analytes to be extracted. Non-linear data analysis and self-organising maps were employed in the data analysis to obtain correlations between the parameters studied, recoveries and relative errors. First, pressurised hot water extraction (PHWE) was combined on-line with liquid chromatography-gas chromatography (LC-GC), and the system was applied to the extraction and analysis of polycyclic aromatic hydrocarbons (PAHs) in sediment. The method is of superior sensitivity compared with the traditional methods, and only a small 10 mg sample was required for analysis. The commercial extraction vessels were replaced by laboratory-made stainless steel vessels because of some problems that arose. The performance of the laboratory-made vessels was comparable to that of the commercial ones. In an investigation of the effect of thermal desorption in PHWE, it was found that at lower temperatures (200ºC and 250ºC) the effect of thermal desorption is smaller than the effect of the solvating property of hot water. At 300ºC, however, thermal desorption is the main mechanism. The effect of the geometry of the extraction vessel on recoveries was studied with five specially constructed extraction vessels. In addition to the extraction vessel geometry, the sediment packing style and the direction of water flow through the vessel were investigated. The geometry of the vessel was found to have only minor effect on the recoveries, and the same was true of the sediment packing style and the direction of water flow through the vessel. These are good results because these parameters do not have to be carefully optimised before the start of extractions. Liquid-liquid extraction (LLE) and solid-phase extraction (SPE) were compared as trapping techniques for PHWE. LLE was more robust than SPE and it provided better recoveries and repeatabilities than did SPE. Problems related to blocking of the Tenax trap and unrepeatable trapping of the analytes were encountered in SPE. Thus, although LLE is more labour intensive, it can be recommended over SPE. The stabilities of the PAHs in aqueous solutions were measured using a batch-type reaction vessel. Degradation was observed at 300ºC even with the shortest heating time. Ketones and quinones and other oxidation products were observed. Although the conditions of the stability studies differed considerably from the extraction conditions in PHWE, the results indicate that the risk of analyte degradation must be taken into account in PHWE. The aqueous solubilities of acenaphthene, anthracene and pyrene were measured, first below and then above the melting point of the analytes. Measurements below the melting point were made to check that the equipment was working, and the results were compared with those obtained earlier. Good agreement was found between the measured and literature values. A new saturation cell was constructed for the solubility measurements above the melting point of the analytes because the flow-through saturation cell could not be used above the melting point. An exponential relationship was found between the solubilities measured for pyrene and anthracene and temperature.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

Theoretical modeling of ionospheric electrodynamics including induction effects

This thesis deals with theoretical modeling of the electrodynamics of auroral ionospheres. In the five research articles forming the main part of the thesis we have concentrated on two main themes: Development of new data-analysis techniques and study of inductive phenomena in the ionospheric electrodynamics. The introductory part of the thesis provides a background for these new results and places them in the wider context of ionospheric research. In this thesis we have developed a new tool (called 1D SECS) for analysing ground based magnetic measurements from a 1-dimensional magnetometer chain (usually aligned in the North-South direction) and a new method for obtaining ionospheric electric field from combined ground based magnetic measurements and estimated ionospheric electric conductance. Both these methods are based on earlier work, but contain important new features: 1D SECS respects the spherical geometry of large scale ionospheric electrojet systems and due to an innovative way of implementing boundary conditions the new method for obtaining electric fields can be applied also at local scale studies. These new calculation methods have been tested using both simulated and real data. The tests indicate that the new methods are more reliable than the previous techniques. Inductive phenomena are intimately related to temporal changes in electric currents. As the large scale ionospheric current systems change relatively slowly, in time scales of several minutes or hours, inductive effects are usually assumed to be negligible. However, during the past ten years, it has been realised that induction can play an important part in some ionospheric phenomena. In this thesis we have studied the role of inductive electric fields and currents in ionospheric electrodynamics. We have formulated the induction problem so that only ionospheric electric parameters are used in the calculations. This is in contrast to previous studies, which require knowledge of the magnetospheric-ionosphere coupling. We have applied our technique to several realistic models of typical auroral phenomena. The results indicate that inductive electric fields and currents are locally important during the most dynamical phenomena (like the westward travelling surge, WTS). In these situations induction may locally contribute up to 20-30% of the total ionospheric electric field and currents. Inductive phenomena do also change the field-aligned currents flowing between the ionosphere and magnetosphere, thus modifying the coupling between the two regions.

Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

Spectral states and accretion geometries in Galactic black hole binaries

Black hole X-ray binaries, binary systems where matter from a companion star is accreted by a stellar mass black hole, thereby releasing enormous amounts of gravitational energy converted into radiation, are seen as strong X-ray sources in the sky. As a black hole can only be detected via its interaction with its surroundings, these binary systems provide important evidence for the existence of black holes. There are now at least twenty cases where the measured mass of the X-ray emitting compact object in a binary exceeds the upper limit for a neutron star, thus inferring the presence of a black hole. These binary systems serve as excellent laboratories not only to study the physics of accretion but also to test predictions of general relativity in strongly curved space time. An understanding of the accretion flow onto these, the most compact objects in our Universe, is therefore of great importance to physics. We are only now slowly beginning to understand the spectra and variability observed in these X-ray sources. During the last decade, a framework has developed that provides an interpretation of the spectral evolution as a function of changes in the physics and geometry of the accretion flow driven by a variable accretion rate. This doctoral thesis presents studies of two black hole binary systems, Cygnus~X-1 and GRS~1915+105, plus the possible black hole candidate Cygnus~X-3, and the results from an attempt to interpret their observed properties within this emerging framework. The main result presented in this thesis is an interpretation of the spectral variability in the enigmatic source Cygnus~X-3, including the nature and accretion geometry of its so-called hard spectral state. The results suggest that the compact object in this source, which has not been uniquely identified as a black hole on the basis of standard mass measurements, is most probably a massive, ~30 Msun, black hole, and thus the most massive black hole observed in a binary in our Galaxy so far. In addition, results concerning a possible observation of limit-cycle variability in the microquasar GRS~1915+105 are presented as well as evidence of `mini-hysteresis' in the extreme hard state of Cygnus X-1.

The British Intervention in South Russia 1918-1920

Soon after the Bolshevik Revolution of 1917, a three-year civil war broke out in Russia. As in many other civil wars, foreign powers intervened in the conflict. Britain played a leading role in this intervention and had a significant effect on the course of the war. Without this intervention on the White side, the superiority of numbers in manpower and weaponry of the Bolsheviks would have quickly overwhelmed their opponents. The aim of this dissertation is to explain the nature and role of the British intervention on the southern, and most decisive, front of the Civil War. The political decision making in London is studied as a background, but the focus of the dissertation is on the actual implementation of the British policy in Russia. The British military mission arrived in South Russia in late 1918, and started to provide General Denikin s White army with ample supplies. General Denikin would have not been able to build his army of more than 200,000 men or to make his operation against Moscow without the British matériel. The British mission also organized the training and equipping of the Russian troops with British weapons. This made the material aid much more effective. Many of the British instructors took part in fighting the Bolsheviks despite the orders of their government. The study is based on primary sources produced by British departments of state and members of the British mission and military units in South Russia. Primary sources from the Whites, including the personal collections of several key figures of the White movement and official records of the Armed Forces of South Russia are also used to give a balanced picture of the course of events. It is possible to draw some general conclusions from the White movement and reasons for their defeat from the study of the British intervention. In purely material terms the British aid placed Denikin s army in a far more favourable position than the Bolsheviks in 1919, but other military defects in the White army were numerous. The White commanders were unimaginative, their military thinking was obsolete, and they were incapable of organizing the logistics of their army. There were also fundamental defects in the morale of the White troops. In addition to all political mistakes of Denikin s movement and a general inability to adjust to the complex situation in Revolutionary Russia, the Whites suffered a clear military defeat. In South Russia the Whites were defeated not because of the lack of British aid, but rather in spite of it.