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.

Konstrukcija mnimosti : K poètike "Voskovoj persony" Ju. Tynjanova

The subject of this work is the poetics of «The Wax Effigy», a short novel or novella by Jurii Tynianov, Russian writer, literary critic, historian of literature and prominent literary theoretician. The plot structure of the novel is based upon a real event, the creation by Bartolomeo Carlo Rastrelli in 1725 of a wax sculpture of the first Russian emperor, Peter the Great. «Construction of the Sham» consists of three chapters, an introduction and a conclusion. Due to the fact that Tynianov was at the same time a prose writer and theoretician of literature it seemed important to consider the reception of his prose and his works on literary theory in relationship to each other. The introduction is devoted to this task. The first chapter is about the history of the creation of the novel and its reception. Tynianov stopped writing one short story in order to write the novel; these two works have some common traits. It seems almost obvious that his work on the first text was a real step toward the creation of the second. In the first story there is an opposition of dead/alive which is semantic prefiguring of a central motif in «The Wax Effigy». An analysis of the reception of the novel demonstrated that almost every critic writing about the novel has described it as nonsense. Critics considered Tynianov's work in terms of «devices» and «content» and could not understand how devices are related to the content of the novel: the novel was thought as a signifier without any signified. Implicitly, critics thought the signified of the novel as a traditional one of the historical novel, as the historiosophical «idea», embodied in the system of literary devices. In this case literature becomes something instrumental, a kind of expression of extraliterary content. In contradistinction to that Tynianov considered literary semantics as an effect of the literary structure. From his point of view the literary sense is immanent to the process of signification accomplished inside the literary text. The second chapter is devoted to a rhetorical analysis of the opposition dead/alive. Tynianov systematically compares both terms of the opposition. As a result of this strategy the wax effigy of the dead emperor becomes «as if» alive and the world of living people «as if» dead. The qualifier «as if» refers to the fact that Tynianov creates an ambiguous semantic system. This rhetoric is related to European Romanticism and his «fantastic literature» (Merimé, Hoffmann, Maupassant etc.). But Tynianov demonstrates a linguistic origin of the strange fantoms created by romantics; he demystifies these idols by parodying the fantastic literature, that is, showing «how it was done». At the same time, the opposition mentioned above refers to his idea of «incongruity» which plays a prominent role in Tynianov s theory but has never been conceptualised. The incongruity is a inner collision of the literary text; from Tynianov's point of view the meaning of the work of literature is always a dynamic collision of semantically heterogeneous elements struggling with each other. In «The Wax Effigy» Tynianov creates a metalevel of the work demonstrating the process of creation of the literary sense. The third chapter is a reconstruction of Tynianov's conception of the historical prose, specifically of the mechanisms by which historical facts are transformed into literary events. Tynianov thought that the task of the historical novelist is to depict his hero as an actor, to demonstrate that as a wearer of many masks he is a creator of appearances, ambiguities. Here, in the «figure of fiction» (Andrei Belyi), the very idea of the historical prose and rhetoric employed in «The Wax Effigy», history and literature meet each other. In his last theoretical work, «On parody» Tynianov writes about the so-called sham structure of parody. In his opinion every parody is a text about other texts and «serious» work which could be read at the same time as a text about «reality». This twofold structure of parody is that of «The Wax Effigy»: that text speaks about ambiguities of the history and about ambiguities of the literary sense, about social reality of the past and - about the working of the literature itself. «The Wax Effigy» is written as a autoreflective text, as an experiment in literary semantics, as a system of literary ambiguities - of hero, rhetoric and the text itself. The meaning of the novel is created not by the embodiment extraliterary idea, but by the process of signification accomplished inside the work of literature. In this sense Tynianov's novel is parody, a break with the tradition of the historical novel preceding «The Wax Effigy».

Olen maa johon tahdot : Timo K. Mukan maailmankuvan poetiikkaa

The dissertation deals with the prose texts of the Finnish writer Timo K. Mukka, renowned for his depictions of his native Lapland. This research concerns the creation of world view in Mukka s prose, which is approached by studying what Mikhail Bakhtin calls generic change. Such genre change is the most characteristic feature of Mukka s prose. His prose is permeated with two genres in particular and changes between them: the ballad-like, archaistic and romantic prose-poem style and naturalistic, even grotesque expressions. In addition, these genres are associated with sublime and grotesque styles so that generic change tends to involve also stylistic changes in Mukka s prose. This study probes the tension-filled interrelationships between the ballad and naturalistic prose by examining the discourse of Mukka s characters. It is shown that these characters invariably find themselves in what Bakhtin calls the chronotope of the threshold; that is, the plots of Mukka s novels and short stories depict situations in which the characters are faced with decisions and deeds that will profoundly impact their lives. The discourse of the threshold affects the characters speech by filling it with dialogical dimensions. This makes their communication ethically loaded and polyphonic. This study is based on Mikhail Bakhtin s theory of the novel and international Bakhtin s studies. I also take into consideration the theoretical developments of Bakhtin s work; for example, the concept of ressentiment, adapted from the Bakhtin scholar Michel André Bernstein, plays an important role. In order to explicate on the psychology of Mukka s characters such as melancholy, abjection, sadism, and taboo I use the concepts familiar from Freudian psychoanalysis. The corpus of my research consists of the following texts: the long prose texts Maa on syntinen laulu. Balladi (1964), Tabu (1965), Täältä jostakin. Romaani (1965), Laulu Sipirjan lapsista. Romaani (1966), Ja kesän heinä kuolee. Kertomus sairaudesta (1968) ja Kyyhky ja unikko (1970) and the short story collections Koiran kuolema (1967) ja Lumen pelko (1970), and with Tabu published short story Sankarihymni , the short story Katkelma laajemmasta laulelmasta from the collection Rakastaa: Kaksitoista novellia rakkaudesta (1965) and also the short stories which were published in various Finnish journals: Yöt (1965), Liisa (1967), Tyttö (1967) ja Näin hetki sitten ketun (1970). I pay particular attention to the novel Maa on syntinen laulu, because it expresses the generic change characteristic of Mukka s world view in a specifically clear and lively way. The dissertation is in Finnish. Key words: Timo K. Mukka, world view, genre change, ballad, naturalism, grotesque realism, Mikhail Bakhtin, dialogism, polyphony, chronotope, sublime, grotesque, ressentiment, Sigmund Freud, melancholy, taboo, abject, sadism, reduced laughter, modern parody.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Yksilöllisyys ja yhteisöllisyys K. E. Nipkowin uskonnonpedagogisen teorian kehityksessä 1960-1990

Individuality and the Community in the Development of K. E. Nipkow's Theory of Religious Education from 1960 to 1990 The purpose of this study was to describe and analyze the development occurred between 1960 and 1990 of the theory of religious education as proposed by K. E. Nipkow, the German Religious Education specialist, from the point of view of individuality and the community. Nipkow's methodological approach of dialectic convergence theory resulted in a dialogue between theological and educational factors, which supported the thirty-year development of Nipkow's models, theoretical foundations, and theory of religious education. Nipkow's doctoral dissertation, published in 1960, deals with individuality in the thinking of Pestalozzi, Humboldt and Schleiermacher. Nipkow regarded individuality as one of the basic concepts of education, which were to be interpreted anew as social and historical situations changed. In the late 1960s Nipkow developed the so-called experiential hermeneutically oriented context model for the needs of religious education. In this model, individuality is expressed in the attention paid to pupils' life situations and the educational reality. The multi-dimensional theoretical framework of religious education in 1975 emphasized supporting identity as a fundamental task of religious education. The concept of individuality was thus given a new form, in accordance with contemporary theories of developmental psychology. Other fundamental tasks, such as the socio-ethical task, the task of critical religious thinking, and that of ecumenical learning, meant a more specific emphasis on the community. It was an outline of a liberating education, which faced the individualistic-existential and social-ecclesiastical challenges of the time with a critical attitude. The further development of the theoretical outline in 1982 continued to uphold the perspectives of both individuality and the community, as Nipkow combined a historical-social dimension with theories of developmental psychology, especially that of life-span research. According to him, the development of the individual and communal life-reality belonged together. The fundamental task of religious education came to be learning to live and believe together. Nipkow transferred the idea of dialogue into inter-generational learning and developed elementarization as a methodology of Religious Education, which takes into account the point of departure of each age group. His theory of educational responsibility in the church (1990) contained the tasks of walking alongside the individual and the renewal of church communities as prerequisites of communicating the Christian faith in an era characterized by multifaceted Christianity. The "geisteswissenschaftliche" school and its concepts (Ger. Individualität; Bildung) were found to be the explanatory factor of the concepts of individuality and the community in the development of Nipkow's theory of religious education. The concept of education employed by Nipkow (Ger. Bildung) implies, on one hand, the individuality, autonomy, freedom and personal responsibility of people of different ages, and on the other hand, the dialogical nature of education in the community facilitated by this concept. Theologically, Nipkow associates himself in his views on individuality and the community with Schleiermacher's understanding of faith, of which openness towards the world was characteristic. The significance of individuality and the community in Nipkow's thinking was, furthermore, deepened by his participation, as a member of working parties, in the educational discussions of the World Council of Churches.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

K -shell diagram and hypersatellite spectra of 4d transition elements

The K-shell diagram (K alpha(1,2) and K beta(1,3)) and hypersatellite (HS) (K-h alpha(1,2)) spectra of Y, Zr, Mo, and Pd have been measured with high energy-resolution using photoexcitation by 90 keV synchrotron radiation. Comparison of the measured and ab initio calculated HS spectra demonstrates the importance of quantum electrodynamical (QED) effects for the HS spectra. Phenomenological fits of the measured spectra by Voigt functions yield accurate values for the shift of the HS from the diagram lines, the splitting of the HS lines, and their intensity ratio. Good agreement with theory was found for all quantities except for the intensity ratio, which is dominated by the intermediacy of the coupling of the angular momenta. The observed deviations imply that our current understanding of the variation of the coupling scheme from LS to jj across the periodic table may require some revision.