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.

Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

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.

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.

Conrad Summenhart's theory of individual rights and its medieval background

This study focuses on the theory of individual rights that the German theologian Conrad Summenhart (1455-1502) explicated in his massive work Opus septipartitum de contractibus pro foro conscientiae et theologico. The central question to be studied is: How does Summenhart understand the concept of an individual right and its immediate implications? The basic premiss of this study is that in Opus septipartitum Summenhart composed a comprehensive theory of individual rights as a contribution to the on-going medieval discourse on rights. With this rationale, the first part of the study concentrates on earlier discussions on rights as the background for Summenhart s theory. Special attention is paid to language in which right was defined in terms of power . In the fourteenth century writers like Hervaeus Natalis and William Ockham maintained that right signifies power by which the right-holder can to use material things licitly. It will also be shown how the attempts to describe what is meant by the term right became more specified and cultivated. Gerson followed the implications that the term power had in natural philosophy and attributed rights to animals and other creatures. To secure right as a normative concept, Gerson utilized the ancient ius suum cuique-principle of justice and introduced a definition in which right was seen as derived from justice. The latter part of this study makes effort to reconstructing Summenhart s theory of individual rights in three sections. The first section clarifies Summenhart s discussion of the right of the individual or the concept of an individual right. Summenhart specified Gerson s description of right as power, taking further use of the language of natural philosophy. In this respect, Summenhart s theory managed to bring an end to a particular continuity of thought that was centered upon a view in which right was understood to signify power to licit action. Perhaps the most significant feature of Summenhart s discussion was the way he explicated the implication of liberty that was present in Gerson s language of rights. Summenhart assimilated libertas with the self-mastery or dominion that in the economic context of discussion took the form of (a moderate) self-ownership. Summenhart discussion also introduced two apparent extensions to Gerson s terminology. First, Summenhart classified right as relation, and second, he equated right with dominion. It is distinctive of Summenhart s view that he took action as the primary determinant of right: Everyone has as much rights or dominion in regard to a thing, as much actions it is licit for him to exercise in regard to the thing. The second section elaborates Summenhart s discussion of the species dominion, which delivered an answer to the question of what kind of rights exist, and clarified thereby the implications of the concept of an individual right. The central feature in Summenhart s discussion was his conscious effort to systematize Gerson s language by combining classifications of dominion into a coherent whole. In this respect, his treatement of the natural dominion is emblematic. Summenhart constructed the concept of natural dominion by making use of the concepts of foundation (founded on a natural gift) and law (according to the natural law). In defining natural dominion as dominion founded on a natural gift, Summenhart attributed natural dominion to animals and even to heavenly bodies. In discussing man s natural dominion, Summenhart pointed out that the natural dominion is not sufficiently identified by its foundation, but requires further specification, which Summenhart finds in the idea that natural dominion is appropriate to the subject according to the natural law. This characterization lead him to treat God s dominion as natural dominion. Partly, this was due to Summenhart s specific understanding of the natural law, which made reasonableness as the primary criterion for the natural dominion at the expense of any metaphysical considerations. The third section clarifies Summenhart s discussion of the property rights defined by the positive human law. By delivering an account on juridical property rights Summenhart connected his philosophical and theological theory on rights to the juridical language of his times, and demonstrated that his own language of rights was compatible with current juridical terminology. Summenhart prepared his discussion of property rights with an account of the justification for private property, which gave private property a direct and strong natural law-based justification. Summenhart s discussion of the four property rights usus, usufructus, proprietas, and possession aimed at delivering a detailed report of the usage of these concepts in juridical discourse. His discussion was characterized by extensive use of the juridical source texts, which was more direct and verbal the more his discussion became entangled with the details of juridical doctrine. At the same time he promoted his own language on rights, especially by applying the idea of right as relation. He also showed recognizable effort towards systematizing juridical language related to property rights.

Spirituality of Finnish peacekeepers

The previous academic research on Finnish peacekeeping has clarified the operative and historical aspects of Finnish peacekeeping lacking the view of an individual who does the actual peacekeeping work. This research is based on the underlying theoretical assumption of human beings possessing different kinds of talents and intelligences creating a holistic entity. In this broad perspective spirituality was explored as an umbrella concept, as a holistic ability or talent, that can be explored as the deepest aspect of defining what it means to be human. The theoretical framework incorporated the concept of an intelligence, which is defined in Gardner's theory of multiple intelligences as the ability to solve problems, or to create products, that are valued within one or more cultural settings (Gardner, 1993, x). The viability of this theory was studied in the sample of Finnish peacekeepers. Spirituality in the theoretical and conceptual horizon was viewed as an extension of Gardner's theory of intelligences as one potential Gardnerian intelligence candidate. In addition to Gardner's theory, spirituality was explored as sensitivity which includes capacities such as sensing awareness, sensing mystery and sensing value (Hay, 1998). Also the practical aspects of spirituality were taken in account as shown in our everyday lives giving us the direction and influencing our social responsibilities and concerns (Bradford, 1995). Spirituality was explored also involving the element of the peacekeepers' community, the element of personal moral orientations and in the domain of religion and coping. The purpose of this research aimed in two dimensions. First, the aim was to outline the intelligence profile and the spiritual sensitivity profile of peacekeepers. Second, the aim was to understand qualitatively the nature of peacekeepers' spirituality The research interests were studied with different kinds of peacekeepers. Applying the mixed methods approach the research was conducted in two phases: first the former SFOR peacekeepers (N=6) were interviewed and the data was analysed. Inspired by the primary findings of these interviews, the data for the case-study of one peacekeeper was collected in co-operation with one former SFOR peacekeeper (N=1). In the second phase the data was collected from KFOR peacekeepers through the quantitative MI-Survey and the spiritual sensitivity survey (N=195). The quantitative method was used to outline the intelligence profile and the spiritual sensitivity profile of peacekeepers (N=195). In the mixed methods approach this method highlighted the general overview of intelligence traits and spiritual sensitivity of peacekeepers. In the mixed methods approach the qualitative method including interviews (N=6) and a case-study of one peacekeeper (N=1) increased subjective, qualitative information of spirituality of peacekeepers. The intelligence profile of peacekeepers highlighted the bodily-kinesthetic and interpersonal dimensions as the practical and social aspects of peacekeepers. Strong inter-item dependencies in the intrapersonal intelligence profile meant that peacekeepers possess a self-reflection and self-knowledge component and they reflect on deep psychological and philosophical issues. Regarding the spiritual sensitivity, peacekeepers found awareness-sensing, mystery-sensing, value-sensing and community-sensing important. The community-sensing emphasised a strong will to advance peace and to help people who are in need: things that are close to the heart of the peacekeepers. These results depicted practicality, being socially capable, and reflecting one's inner world as essential to peacekeepers. Moreover, spirituality as peacekeepers' moral endeavour became clearer because the sub-model of their community-sensing described morally charged destinations: advancing peace and helping people in need. In the qualitative findings peacekeepers articulated justice orientation and rule-following characterising the nature of peacekeepers' moral attitude and moral call (Kohlberg, 1969). An ethic of care (Gilligan, 1982) describes mainly female moral orientation, but the findings revealed that an ethic of care is also an important agent supporting strongly male peacekeepers in their aim to carry out qualitatively good peacekeeping work. The moral endeavour was voiced, when the role of religion in coping meant the assessment of the a way of life, a way of conduct, a way of being truthful to one's own values in confusing surroundings. The practical level of spiritual and religious contemplation was voiced as morally charged inner motivation to fulfil one's duties and at the same time to cope with various peacekeeping challenges. The results of different data sets were combined and interpreted as the moral endeavour, which characterises peacekeepers' spirituality. As the combining result, the perspective of peacekeepers' spirituality is considered moral or at least morally charged.

Pyhä Teresa ja keskiajan mystiikan teoriat

In my dissertation I have studied St Teresa (1515-1582) in the light of medieval mystical theories. I have two main levels in my research: historical and theological. On the historical level I study St Teresa s personal history in the context of her family and the Spanish society. On the theological level I study both St Teresa s mysticism and her religious experience in the light of medieval mysticism. St Teresa wrote a book called Life , which is her narrative autobiography and story about her mystical spiritual formation. She reflected herself through biblical texts interpreting them in the course of the biblical hermeneutics like allegory, typology, tropology and anagogy. In addition to that she read others life stories from her period of time, but reflected herself only slightly through the sociological point of view. She used irony as a means to gain acceptance to her authority and motive to write. Her position has been described as a double bind because of writing at the request of educated men and to the non-educated women as she herself was uneducated. She used irony as a means to achieve valuation to women, to gain negative attributes connected to them and to gain authority to teach them mystical spirituality, the Bible and prayer. In this ironic tendency she was a feminist writer. In order to understand medieval mysticism I have written in the first chapter a review of the main trends in medieval mysticism in connection with the classical emotional theories. Two medieval mystical theories show an important role in St Teresa s mysticism. One is love mysticism and the other is the three partite way of mysticism (purification, illumination and union). The classic-philosophical emotional theories play a role in both patterns. The theory of love mysticism St Teresa interpreted in the traditional way stressing the spiritual meaning of love in connexion with God and neighbors. Love is an emotion, which is bound with other emotions, but all objects of love don t strengthen spiritual love. In the three partite way of mysticism purification means to find biblical values in life and to practice meditative self-knowledge theologically interpreted. In illumination human understanding has to be illuminated by God and united to mystical knowledge from God. St Teresa considered illumination a way to learn things. Illumination has also psychological aspects like recognition of many trials and pains, which come from life on earth. Theologically interpreted in illumination one should die to oneself, let oneself be transformed and renewed by God. I have also written a review of the modern philosophical discussion on personal identity where memory and mental experiences are important creators of personal identity. St Teresa bound medieval mystical teaching together with her personal religious experience. Her personal identity is by its character based on her narrative life story where mental experiences play important role. Previous researchers have labelled St Teresa as an ecstatic person whose experiences produced ecstatic phenomena to the mysticism. These phenomena combined with visions have in one respect made of her a person who has brought physical and visionary tendencies to theology. In spite of that she also represents a modern tendency trying to give words to experiences, which at first seem to be exceptional and extreme and which are easily interpreted as one-sided either physical or sexual or unsaid. In other respect I have stressed the personality of St Teresa that was represented as both strong and weak. The strong personality for her is demonstrated by religious faith and in its practice. The weak personality was for her a natural personal identity. St Teresa saw a unifying aspect in almost all. Firstly, her mysticism was aimed towards union with God and secondly, the unifying aspects and common rules in human relations in community life were central. Union with God is based on the fact that in a soul God is living in its centre, where God is present in the Trinitarian way. The picture of God in ourselves is a mirror but to get to know God better is to recognize his/her presence in us. When the soul recognizes itself as a dwelling place of God, it knows itself as God knows him/herself. There is equality between God and the soul. To be a Christian means to participate in God in his Trinitarian being. The participation to God is a process of divinization that puts a person into transformation, change and renewal. The unitive aspect concludes also knowledge of opposites between experience of community and solitude as well as community and separateness. As a founder of monasteries St Teresa practiced theology of poverty. She renewed the monastic life founding a rule called discalced that stressed ascetic tendencies. Supporters of her work were after the difficulties in the beginning both society and churchly leaders. She wrote about the monasteries including in her description at times seriousness at times humor and irony. Her stories are said to be picaresque histories that contain stories of ordinary laymen and many unexpected occasions. She exercised a kind of Bakhtinian dialogue in her letters. St Teresa stressed the virtues like sacrifice, determination and courage in the monastic life. Most of what she taught of virtues is based on biblical spirituality but there are also psychological tendencies in her writings. The theological pedagogical advice is mixed with psychology, but she herself made no distinction between different aspects in her teaching. To understand St Teresa and her mysticism is to recognize that she mixes her personal religious experience and mysticism, which widens mysticism to religious experience in a new way, although this corresponds also the very definition of mysticism. St Teresa concentrated on mental-spiritual experiences and the aim of her mystical teaching was to produce a human mind well cured like a garden that has God as its gardener.

Imagination and diversity in the philosophy of Hobbes

This dissertation is a study of some aspects of theoretical philosophy of the early modern thinker Thomas Hobbes (1588-1679). The focal point of the work is Hobbes s conception of imagination, which is discussed from both a systematic and a historical point of view, as well as in the light of contemporary scholarship. I argue that though there are significant similarities between the view of Hobbes and that of his predecessors, he gives a novel theory of imagination, which clarifies not only early modern discussions on human nature, knowledge, science, and literary criticism, but above all his own versatile philosophy. The prologue of the dissertation introduces methodological principles and gives critical remarks on the standard view of Hobbes. In Chapter II, I discuss the prominent theories of imagination before Hobbes and link them to his account. I argue that though Hobbes adopted the Aristotelian framework, his view is not reduced to it, as he borrows from various sources, for instance, from the Stoics and from Renaissance thought. Chapters III and IV form the psychological part of the work. In the Chapter III I argue that imagination, not sense, is central in the basic cognitive operations of the mind and that imagination has a decisive role in Hobbes s theory of motivation. The Chapter IV concentrates on various questions of Hobbes s philosophy of language. The chapter ends with a defence of a less naturalistic reading of Hobbes s theory of human nature. Chapters V and VI form the epistemological part of the work. I suggest, contrary to what has been recently claimed, that though Hobbes s ideas of good literary style do have a point of contact with his philosophy (e.g. the psychology of creative process), his ideas in the field are independent of his project of demonstrative political science. Instead I argue that the novelty of his major political work, Leviathan (1651), is based on a new theory of knowledge which he continued to develop in the post-Leviathan works. Chapter VII seeks to connect the more theoretical conclusions of Chapters V and VI to Hobbes's idea(l) of science as well as to his philosophical practice. On the basis of Hobbes s own writings as well as some historical examinations, I argue that method is not an apt way to conceptualise Hobbes s philosophical practice. Contemporary readings of Hobbes s theory of science are critically discussed and the chapter ends with an analysis of Hobbes s actual argumentation. In addition to the concluding remarks, the epilogue suggest three things: first, imagination is central when trying to understand Hobbes s versatile philosophy; second, that it is misleading to depict Hobbes as a simple materialist, mechanist, and empiricist; and, third, that in terms of imagination his influence on early modern thought has not been fully appreciated.

Essays in Philosophical Moral Psychology

The dissertation consists of four essays and a comprehensive introduction that discusses the topics, methods, and most prominent theories of philosophical moral psychology. I distinguish three main questions: What are the essential features of moral thinking? What are the psychological conditions of moral responsibility? And finally, what are the consequences of empirical facts about human nature to normative ethics? Each of the three last articles focuses on one of these issues. The first essay and part of the introduction are dedicated to methodological questions, in particular the relationship between empirical (social) psychology and philosophy. I reject recent attempts to understand the nature of morality on the basis of empirical research. One characteristic feature of moral thinking is its practical clout: if we regard an action as morally wrong, we either refrain from doing it even against our desires and interests, or else feel shame or guilt. Moral views seem to have a conceptual connection to motivation and emotions – roughly speaking, we can’t conceive of someone genuinely disapproving an action, but nonetheless doing it without any inner motivational conflict or regret. This conceptual thesis in moral psychology is called (judgment) internalism. It implies, among other things, that psychopaths cannot make moral judgments to the extent that they are incapable of corresponding motivation and emotion, even if they might say largely the words we would expect. Is internalism true? Recently, there has been an explosion of interest in so-called experimental philosophy, which is a methodological view according to which claims about conceptual truths that appeal to our intuitions should be tested by way of surveys presented to ordinary language users. One experimental result is that the majority of people are willing to grant that psychopaths make moral judgments, which challenges internalism. In the first article, ‘The Rise and Fall of Experimental Philosophy’, I argue that these results pose no real threat to internalism, since experimental philosophy is based on a too simple conception of the relationship between language use and concepts. Only the reactions of competent users in pragmatically neutral and otherwise conducive circumstances yield evidence about conceptual truths, and such robust intuitions remain inaccessible to surveys for reasons of principle. The epistemology of folk concepts must still be based on Socratic dialogue and critical reflection, whose character and authority I discuss at the end of the paper. The internal connection between moral judgment and motivation led many metaethicists in the past century to believe along Humean lines that judgment itself consists in a pro-attitude rather than a belief. This expressivist view, as it is called these days, has far-reaching consequences in metaethics. In the second essay I argue that perhaps the most sophisticated form of contemporary expressivism, Allan Gibbard’s norm-expressivism, according to which moral judgments are decisions or contingency plans, is implausible from the perspective of the theory of action. In certain circumstances it is possible to think that something is morally required of one without deciding to do so. Morality is not a matter of the will. Instead, I sketch on the basis of Robert Brandom’s inferentialist semantics a weak form of judgment internalism, according to which the content of moral judgment is determined by a commitment to a particular kind of practical reasoning. The last two essays in the dissertation emphasize the role of mutual recognition in the development and maintenance of responsible and autonomous moral agency. I defend a compatibilist view of autonomy, according to which agents who are unable to recognize right and wrong or act accordingly are not responsible for their actions – it is not fair to praise or blame them, since they lacked the relevant capacity to do otherwise. Conversely, autonomy demands an ability to recognize reasons and act on them. But as a long tradition in German moral philosophy whose best-known contemporary representative is Axel Honneth has it, both being aware of reasons and acting on them requires also the right sort of higher-order attitudes toward the self. Without self-respect and self-confidence we remain at the mercy of external pressures, even if we have the necessary normative competence. These attitudes toward the self, in turn, are formed through mutual recognition – we value ourselves when those who we value value us. Thus, standing in the right sort of relations of recognition is indirectly necessary for autonomy and moral responsibility. Recognition and valuing are concretely manifest in actions and institutions, whose practices make possible participation on an equal footing. Seeing this opens the way for a kind of normative social criticism that is grounded in the value of freedom and automomy, but is not limited to defending negative rights. It thus offers a new way to bridge the gap between liberalism and communitarianism.