Confucian education and enlightenment for the masses in the manner of Guwen guanzhi

The topic of this study is the most renowned anthology of essays written in Literary Chinese, Guwen guanzhi, compiled and edited by Wu Chengquan (Chucai) and Wu Dazhi (Diaohou), and first published during the Qing dynasty, in 1695. Because of the low social standing of the compilers, their anthology remained outside the recommended study materials produced by members of the established literati and used for preparing students in the imperial civil-service examinations. However, since the end of the imperial era, Guwen guanzhi has risen to a position as the classical anthology par excellence. Today it is widely used as required or supplementary reading material of Literary Chinese in middle-schools both in Mainland China and on Taiwan. The goal of this study is to explain the persistent longevity of the anthology. So far, Guwen guanzhi has not been a topic of any published academic study, and the opinions expressed on it in various sources are widely discrepant. Through a comparative study with a dozen classical Chinese anthologies in use during the early Qing dynasty, this study reveals the extent to which the compilers of Guwen guanzhi modelled their work after other selections. Altogether 86 % of the texts in Guwen guanzhi originate from another Qing era anthology, Guwen xiyi, often copied character by character. However, the notes and commentaries are all different. Concentrating on the special characteristics unique to Guwen guanzhi—the commentaries and certain peculiarities in the selection of texts—this study then discusses the possible reasons for the popularity of Guwen guanzhi over the competing readers during the Qing era. Most remarkably, Guwen guanzhi put in practise the equalitarian, educational ideals of the Ming philosopher Wang Shouren (Yangming). Thus Guwen guanzhi suited the self-enlightenment needs of the ”subordinate classes”, in particular the rising middle-class comprised mainly of merchants. The lack of moral teleology, together with the compact size, relative comprehensiveness of the selection and good notes and comments, have made Guwen guanzhi well suited for the new society since the abolition of the imperial examination system. Through a content analysis, based on a sample of the texts, this study measures the relative emphasis on centralism and localism (both in concrete and spiritual terms) expressed in the texts of Guwen guanzhi. The analysis shows that the texts manifest some bias towards emphasising innate virtue on the expense of state-defined moral. This may reflect hidden critique towards intellectual oppression by the centralised imperial rule. During the early decades of the Qing era, such critique was often linked to Ming-loyalism. Finally, this study concludes that the kind of ”spiritual localism” that Guwen guanzhi manifests gives it the potential to undermine monolithic orthodoxy even in today’s Chinese societies. This study has progressed hand in hand with the translation of a selection of texts from Guwen guanzhi into Finnish, published by Gaudeamus Helsinki University Press: Jadekasvot – Valittuja tarinoita Kiinan muinaisajoilta (2005), Jadelähde – Valittuja kirjoituksia Kiinan keskiajalta (2007) and Jadepeili – Valittuja kirjoituksia keisarillisen Kiinan kulta-ajoilta (2008). All translations are critical editions, complete with extensive notation. The trilogy is the first comprehensive translation based on Guwen guanzhi in a European language.

Tyhjyyden reunalla, helvetin kannella : Yhtenäisen subjektin hajoaminen Saul Bellowin Herzogissa

The study shows that the reading paradigm derived from codes Roland Barthes presented in his S/Z exposes the postmodernic nature of Saul Bellow s Herzog (1964), and embodies in which way the novel is organized as a radical rewriting of modernism and its ideology. I explore how solid, compact and modernistic subject breaks down at every level of Herzog s narration. I actually argue that the heteroglossic novel is representative of an early American postmodern movement in literaure, and it should by no means be dissected narrowly as a realistic or naturalistic novel at all. The intertextual code verifies that the interpretation of Herzog remains inadequate if one doesn t take account of the novel s significant intertextual references to other texts. In fact, even the mind of Moses Herzog, the protagonist of the novel, is a mosaic of citations. It emerges from the dissertation that the figure of Don Quixote follows Moses Herzog as an ambiguous shadow while the professor of history struggles with his anxiety and anguish, and travels in a mentally confused state around the U.S. for five days in the early 1960s, encountering the impending atmosphere of transition as the country is on the threshold of a significant cultural and social change. There is a strong necessity for updated interpretation of Herzog partly because its centrality to Bellow s own career as a writer but mainly because it has been previously read trough modernistic lenses. I shall try to proof in my study that American Jewish Saul Bellow s (1915 2005) Herzog escapes any kind of simple, elemental or essentialistic construction or reading and in real terms it doesn t offer any comprehensive, total or coherent solution or system for those philosophical doctrines it criticizes and makes fun of. The philosophical, conceptual and cultural substance of Herzog does not constitute an independent or autonomous theoretical tract which would have a life of its own. Altough the novel reflects the continuity of Bellow s writing it is clearly some kind of conscious or unconscious experiment during his long career as a writer. He hasn t been so radical before or since the publication of Herzog. It is unarguably his most postmodern novel.The reading paradigm based on specific codes demonstrates how deep into the basic questions of his personal life and existence itself he must dive in oder to find his many ways towards authentic or primordial self in fragmented and shattered world which is constanly rewritten and which makes human being a tourist of his own life. In that ongoing process the protagonist has to accept the ultimate plurality of his mind and self. He must confront that the modernistic definition of identity as a solid, monolithic and stable entity has broken down into different, inconsistent and even contradictory possibilities of identification. Moses Herzog embodies obviously Stuart Hall s description of the postmodern self his identity has turn into a movable feast.

Teacher Education, School Effectiveness and Improvement : A Study of Academic and Professional Qualification on Teachers' Job Effectiveness in Nigerian Secondary Schools

This academic work begins with a compact presentation of the general background to the study, which also includes an autobiography for the interest in this research. The presentation provides readers who know little of the topic of this research and of the structure of the educational system as well as of the value given to education in Nigeria. It further concentrates on the dynamic interplay of the effect of academic and professional qualification and teachers' job effectiveness in secondary schools in Nigeria in particular, and in Africa in general. The aim of this study is to produce a systematic analysis and rich theoretical and empirical description of teachers' teaching competencies. The theoretical part comprises a comprehensive literature review that focuses on research conducted in the areas of academic and professional qualification and teachers' job effectiveness, teaching competencies, and the role of teacher education with particular emphasis on school effectiveness and improvement. This research benefits greatly from the functionalist conception of education, which is built upon two emphases: the application of the scientific method to the objective social world, and the use of an analogy between the individual 'organism' and 'society'. To this end, it offers us an opportunity to define terms systematically and to view problems as always being interrelated with other components of society. The empirical part involves describing and interpreting what educational objectives can be achieved with the help of teachers' teaching competencies in close connection to educational planning, teacher training and development, and achieving them without waste. The data used in this study were collected between 2002 and 2003 from teachers, principals, supervisors of education from the Ministry of Education and Post Primary Schools Board in the Rivers State of Nigeria (N=300). The data were collected from interviews, documents, observation, and questionnaires and were analyzed using both qualitative and quantitative methods to strengthen the validity of the findings. The data collected were analyzed to answer the specific research questions and hypotheses posited in this study. The data analysis involved the use of multiple statistical procedures: Percentages Mean Point Value, T-test of Significance, One-Way Analysis of Variance (ANOVA), and Cross Tabulation. The results obtained from the data analysis show that teachers require professional knowledge and professional teaching skills, as well as a broad base of general knowledge (e.g., morality, service, cultural capital, institutional survey). Above all, in order to carry out instructional processes effectively, teachers should be both academically and professionally trained. This study revealed that teachers are not however expected to have an extraordinary memory, but rather looked upon as persons capable of thinking in the right direction. This study may provide a solution to the problem of teacher education and school effectiveness in Nigeria. For this reason, I offer this treatise to anyone seriously committed in improving schools in developing countries in general and in Nigeria in particular to improve the lives of all its citizens. In particular, I write this to encourage educational planners, education policy makers, curriculum developers, principals, teachers, and students of education interested in empirical information and methods to conceptualize the issue this study has raised and to provide them with useful suggestions to help them improve secondary schooling in Nigeria. Though, multiple audiences exist for any text. For this reason, I trust that the academic community will find this piece of work a useful addition to the existing literature on school effectiveness and school improvement. Through integrating concepts from a number of disciplines, I aim to describe as holistic a representation as space could allow of the components of school effectiveness and quality improvement. A new perspective on teachers' professional competencies, which not only take into consideration the unique characteristics of the variables used in this study, but also recommend their environmental and cultural derivation. In addition, researchers should focus their attention on the ways in which both professional and non-professional teachers construct and apply their methodological competencies, such as their grouping procedures and behaviors to the schooling of students. Keywords: Professional Training, Academic Training, Professionally Qualified, Academically Qualified, Professional Qualification, Academic Qualification, Job Effectiveness, Job Efficiency, Educational Planning, Teacher Training and Development, Nigeria.

Occupational determinants of pancreatic cancer

Objective and background. Tobacco smoking, pancreatitis and diabetes mellitus are the only known causes of pancreatic cancer, leaving ample room for yet unidentified determinants. This is an empirical study on a Finnish data on occupational exposures and pancreatic cancer risk, and a non-Bayesian and a hierarchical Bayesian meta-analysis of data on occupational factors and pancreatic cancer. Methods. The case-control study analyzed 595 incident cases of pancreatic cancer and 1,622 controls of stomach, colon, and rectum cancer, diagnosed 1984-1987 and known to be dead by 1990 in Finland. The next-of-kin responded to a mail questionnaire on job and medical histories and lifestyles. Meta-analysis of occupational risk factors of pancreatic cancer started off with 1,903 identified studies. The analyses were based on different subsets of that database. Five epidemiologists examined the reports and extracted the pertinent data using a standardized extraction form that covered 20 study descriptors and the relevant relative risk estimates. Random effects meta-analyses were applied for 23 chemical agents. In addition, hierarchical Bayesian models for meta-analysis were applied to the occupational data of 27 job titles using job exposure matrix as a link matrix and estimating the relative risks of pancreatic cancer associated with nine occupational agents. Results. In the case-control study, logistic regressions revealed excess risks of pancreatic cancer associated with occupational exposures to ionizing radiation, nonchlorinated solvents, and pesticides. Chlorinated hydrocarbon solvents and related compounds, used mainly in metal degreasing and dry cleaning, are emerging as likely risk factors of pancreatic cancer in the non-Bayesian and the hierarchical Bayesian meta-analysis. Consistent excess risk was found for insecticides, and a high excess for nickel and nickel compounds in the random effects meta-analysis but not in the hierarchical Bayesian meta-analysis. Conclusions. In this study occupational exposure to chlorinated hydrocarbon solvents and related compounds and insecticides increase risk of pancreatic cancer. Hierarchical Bayesian meta-analysis is applicable when studies addressing the agent(s) under study are lacking or very few, but several studies address job titles with potential exposure to these agents. A job-exposure matrix or a formal expert assessment system is necessary in this situation.

Heterotrophic Bacteria Associated with Cyanobacteria in Recreational and Drinking Water

Cyanobacterial mass occurrences, also known as water blooms, have been associated with adverse health effects of both humans and animals. They can also be a burden to drinking water treatment facilities. Risk assessments of the blooms have generally focused on the cyanobacteria themselves and their toxins. However, heterotrophic bacteria thriving among cyanobacteria may also be responsible for many of the adverse health effects, but their role as the etiological agents of these health problems is poorly known. In addition, studies on the water purification efficiency of operating water treatment plants during cyanobacterial mass occurrences in their water sources are rare. In the present study, over 600 heterotrophic bacterial strains were isolated from natural freshwater, brackish water or from treated drinking water. The sampling sites were selected as having frequent cyanobacterial occurrences in the water bodies or in the water sources of the drinking water treatment plants. In addition, samples were taken from sites where cyanobacterial water blooms were surmised to have caused human health problems. The isolated strains represented bacteria from 57 different genera of the Gamma-, Alpha- or Betaproteobacteria, Actinobacteria, Flavobacteria, Sphingobacteria, Bacilli and Deinococci classes, based on their partial 16S rRNA sequences. Several isolates had no close relatives among previously isolated bacteria or cloned 16S rRNA genes of uncultivated bacteria. The results show that water blooms are associated with a diverse community of cultivable heterotrophic bacteria. Chosen subsets of the isolated strains were analysed for features such as their virulence gene content and possible effect on cyanobacterial growth. Of the putatively pathogenic haemolytic strains isolated in the study, the majority represented the genus Aeromonas. Therefore, the Aeromonas spp. strains isolated from water samples associated with adverse health effects were screened for the virulence gene types encoding for enterotoxins (ast, alt and act/aerA/hlyA), flagellin subunits (flaA/flaB), lipase (lip/pla/lipH3/alp-1) and elastase (ahyB) by PCR. The majority (90%) of the Aeromonas strains included one or more of the six screened Aeromonas virulence gene types. The most common gene type was act, which was present in 77% of the strains. The fla, ahyB and lip genes were present in 30 37% of the strains. The prevalence of the virulence genes implies that the Aeromonas may be a factor in some of the cyanobacterial associated health problems. Of the 183 isolated bacterial strains that were studied for possible effects on cyanobacterial growth, the majority (60%) either enhanced or inhibited growth of cyanobacteria. In most cases, they enhanced the growth, which implies mutualistic interactions. The results indicate that the heterotrophic bacteria have a role in the rise and fall of the cyanobacterial water blooms. The genetic and phenotypic characteristics and the ability to degrade cyanobacterial hepatotoxins of 13 previously isolated Betaproteobacteria strains, were also studied. The strains originated from Finnish lakes with frequent cyanobacterial occurrence. Tested strains degraded microcystins -LR and -YR and nodularin. The strains could not be assigned to any described bacterial genus or species based on their genetic or phenotypic features. On the basis of their characteristics a new genus and species Paucibacter toxinivorans was proposed for them. The water purification efficiency of the drinking water treatment processes during cyanobacterial water bloom in water source was assessed at an operating surface water treatment plant. Large phytoplankton, cyanobacterial hepatotoxins, endotoxins and cultivable heterotrophic bacteria were efficiently reduced to low concentrations, often below the detection limits. In contrast, small planktonic cells, including also possible bacterial cells, regularly passed though the water treatment. The passing cells may contribute to biofilm formation within the water distribution system, and therefore lower the obtained drinking water quality. The bacterial strains of this study offer a rich source of isolated strains for examining interactions between cyanobacteria and the heterotrophic bacteria associated with them. The degraders of cyanobacterial hepatotoxins could perhaps be utilized to assist the removal of the hepatotoxins during water treatment, whereas inhibitors of cyanobacterial growth might be useful in controlling cyanobacterial water blooms. The putative pathogenicity of the strains suggests that the health risk assessment of the cyanobacterial blooms should also cover the heterotrophic bacteria.

Geological factors affecting on after-use of Finnish cut-over peatlands : with implications on the carbon accumulation

In Finland, peat harvesting sites are utilized down almost to the mineral soil. In this situation the properties of mineral subsoil are likely to have considerable influence on the suitability for the various after-use forms. The aims of this study were to recognize the chemical and physical properties of mineral subsoils possibly limiting the after-use of cut-over peatlands, to define a minimum practice for mineral subsoil studies and to describe the role of different geological areas. The future percentages of the different after-use forms were predicted, which made it possible to predict also carbon accumulation in this future situation. Mineral subsoils of 54 different peat production areas were studied. Their general features and grain size distribution was analysed. Other general items studied were pH, electrical conductivity, organic matter, water soluble nutrients (P, NO3-N, NH4-N, S and Fe) and exchangeable nutrients (Ca, Mg and K). In some cases also other elements were analysed. In an additional case study carbon accumulation effectiveness before the intervention was evaluated on three sites in Oulu area (representing sites typically considered for peat production). Areas with relatively sulphur rich mineral subsoil and pool-forming areas with very fine and compact mineral subsoil together covered approximately 1/5 of all areas. These areas were unsuitable for commercial use. They were recommended for example for mire regeneration. Another approximate 1/5 of the areas included very coarse or very fine sediments. Commercial use of these areas would demand special techniques - like using the remaining peat layer for compensating properties missing from the mineral subsoil. One after-use form was seldom suitable for one whole released peat production area. Three typical distribution patterns (models) of different mineral subsoils within individual peatlands were found. 57 % of studied cut-over peatlands were well suited for forestry. In a conservative calculation 26% of the areas were clearly suitable for agriculture, horticulture or energy crop production. If till without large boulders was included, the percentage of areas suitable to field crop production would be 42 %. 9-14 % of all areas were well suitable for mire regeneration or bird sanctuaries, but all areas were considered possible for mire regeneration with correct techniques. Also another 11 % was recommended for mire regeneration to avoid disturbing the mineral subsoil, so total 20-25 % of the areas would be used for rewetting. High sulphur concentrations and acidity were typical to the areas below the highest shoreline of the ancient Litorina sea and Lake Ladoga Bothnian Bay zone. Also differences related to nutrition were detected. In coarse sediments natural nutrient concentration was clearly higher in Lake Ladoga Bothnian Bay zone and in the areas of Svecokarelian schists and gneisses, than in Granitoid area of central Finland and in Archaean gneiss areas. Based on this study the recommended minimum analysis for after-use planning was for pH, sulphur content and fine material (<0.06 mm) percentage. Nutrition capacity could be analysed using the natural concentrations of calcium, magnesium and potassium. Carbon accumulation scenarios were developed based on the land-use predictions. These scenarios were calculated for areas in peat production and the areas released from peat production (59300 ha + 15 671 ha). Carbon accumulation of the scenarios varied between 0.074 and 0.152 million t C a-1. In the three peatlands considered for peat production the long term carbon accumulation rates varied between 13 and 24 g C m-2 a-1. The natural annual carbon accumulation had been decreasing towards the time of possible intervention.

Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Luonnollisten lukujen joukon määriteltävyys funktioalgebroissa

Let X be a topological space and K the real algebra of the reals, the complex numbers, the quaternions, or the octonions. The functions form X to K form an algebra T(X,K) with pointwise addition and multiplication. We study first-order definability of the constant function set N' corresponding to the set of the naturals in certain subalgebras of T(X,K). In the vocabulary the symbols Constant, +, *, 0', and 1' are used, where Constant denotes the predicate defining the constants, and 0' and 1' denote the constant functions with values 0 and 1 respectively. The most important result is the following. Let X be a topological space, K the real algebra of the reals, the compelex numbers, the quaternions, or the octonions, and R a subalgebra of the algebra of all functions from X to K containing all constants. Then N' is definable in , if at least one of the following conditions is true. (1) The algebra R is a subalgebra of the algebra of all continuous functions containing a piecewise open mapping from X to K. (2) The space X is sigma-compact, and R is a subalgebra of the algebra of all continuous functions containing a function whose range contains a nonempty open set of K. (3) The algebra K is the set of reals or the complex numbers, and R contains a piecewise open mapping from X to K and does not contain an everywhere unbounded function. (4) The algebra R contains a piecewise open mapping from X to the set of the reals and function whose range contains a nonempty open subset of K. Furthermore R does not contain an everywhere unbounded function.

Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

Dirichlet problem at infinity for p-harmonic functions on negatively curved spaces

The object of this dissertation is to study globally defined bounded p-harmonic functions on Cartan-Hadamard manifolds and Gromov hyperbolic metric measure spaces. Such functions are constructed by solving the so called Dirichlet problem at infinity. This problem is to find a p-harmonic function on the space that extends continuously to the boundary at inifinity and obtains given boundary values there. The dissertation consists of an overview and three published research articles. In the first article the Dirichlet problem at infinity is considered for more general A-harmonic functions on Cartan-Hadamard manifolds. In the special case of two dimensions the Dirichlet problem at infinity is solved by only assuming that the sectional curvature has a certain upper bound. A sharpness result is proved for this upper bound. In the second article the Dirichlet problem at infinity is solved for p-harmonic functions on Cartan-Hadamard manifolds under the assumption that the sectional curvature is bounded outside a compact set from above and from below by functions that depend on the distance to a fixed point. The curvature bounds allow examples of quadratic decay and examples of exponential growth. In the final article a generalization of the Dirichlet problem at infinity for p-harmonic functions is considered on Gromov hyperbolic metric measure spaces. Existence and uniqueness results are proved and Cartan-Hadamard manifolds are considered as an application.

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.

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.