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.

Matrix Decomposition Methods for Data Mining : Computational Complexity and Algorithms

Matrix decompositions, where a given matrix is represented as a product of two other matrices, are regularly used in data mining. Most matrix decompositions have their roots in linear algebra, but the needs of data mining are not always those of linear algebra. In data mining one needs to have results that are interpretable -- and what is considered interpretable in data mining can be very different to what is considered interpretable in linear algebra. --- The purpose of this thesis is to study matrix decompositions that directly address the issue of interpretability. An example is a decomposition of binary matrices where the factor matrices are assumed to be binary and the matrix multiplication is Boolean. The restriction to binary factor matrices increases interpretability -- factor matrices are of the same type as the original matrix -- and allows the use of Boolean matrix multiplication, which is often more intuitive than normal matrix multiplication with binary matrices. Also several other decomposition methods are described, and the computational complexity of computing them is studied together with the hardness of approximating the related optimization problems. Based on these studies, algorithms for constructing the decompositions are proposed. Constructing the decompositions turns out to be computationally hard, and the proposed algorithms are mostly based on various heuristics. Nevertheless, the algorithms are shown to be capable of finding good results in empirical experiments conducted with both synthetic and real-world data.

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.

The Nature, Origins, and Consequences of Finnish Shame-Proneness: A Grounded Theory Study

Although shame is a universal human emotion and is one of the most difficult emotions to overcome, its origins and nature as well as its effects on psychosocial functioning are not well understood or defined. While psychological and spiritual counselors are aware of the effects and consequences of shame for an individual s internal well-being and social life, shame is often still considered a taboo topic and is not given adequate attention. This study aims to explain the developmental process and effects of shame and shame-proneness for individuals and provide tools for practitioners to work more effectively with their clients who struggle with shame. This study presents the empirical foundation for a grounded theory that describes and explains the nature, origins, and consequences of shame-proneness. The study focused on Finnish participants childhood, adolescence and adulthood experiences and why they developed shame-proneness, what it meant for them as children and adolescents and what it meant for them as adults. The data collection phase of this study began in 2000. The participants were recruited through advertisements in local and country-wide newspapers and magazines. Altogether 325 people responded to the advertisements by sending an essay concerning their shame and guilt experiences. For the present study, 135 essays were selected and from those who sent an essay 19 were selected for in-depth interviews. In addition to essays and interviews, participants personal notebooks and childhood hospital and medical reports as well as their scores on the Internalized Shame Scale were analyzed. The development of shame-proneness and significant experiences and events during childhood and adolescence (e.g., health, parenting and parents behavior, humiliation, bullying, neglect, maltreatment and abuse) are discussed and the connections of shame-proneness to psychological concepts such as self-esteem, attachment, perfectionism, narcissism, submissiveness, pleasing others, heightened interpersonal subjectivity, and codependence are explained. Relationships and effects of shame-proneness on guilt, spirituality, temperament, coping strategies, defenses, personality formation and psychological health are also explicated. In addition, shame expressions and the development of shame triggers as well as internalized and externalized shame are clarified. These connections and developments are represented by the core category lack of gaining love, validation and protection as the authentic self. The conclusions drawn from the study include a categorization of shame-prone Finnish people according to their childhood and adolescent experiences and the characteristics of their shame-proneness and personality. Implications for psychological and spiritual counseling are also discussed. Key words: shame, internalized shame, external shame, shame development, shame triggers, guilt, self-esteem, attachment, narcissism, perfectionism, submissiveness, codependence, childhood neglect, childhood abuse, childhood maltreatment, emotional abuse, sexual abuse, spiritual abuse, psychological well-being

Co-Signaling by Neurotrophic Factors and the Extracellular Matrix for Axonal Growth and Neuronal Survival

Neurotrophic factors (NTFs) and the extracellular matrix (ECM) are important regulators of axonal growth and neuronal survival in mammalian nervous system. Understanding of the mechanisms of this regulation is crucial for the development of posttraumatic therapies and drug intervention in the injured nervous system. NTFs act as soluble, target-derived extracellular regulatory molecules for a wide range of physiological functions including axonal guidance and the regulation of programmed cell death in the nervous system. The ECM determines cell adhesion and regulates multiple physiological functions via short range cell-matrix interactions. The present work focuses on the mechanisms of the action of NTFs and the ECM on axonal growth and survival of cultured sensory neurons from dorsal root ganglia (DRG). We first examined signaling mechanisms of the action of the glial cell line-derived neurotrophic factor (GDNF) family ligands (GFLs) on axonal growth. GDNF, neurturin (NRTN) and artemin (ART) but not persephin (PSPN) promoted axonal initiation in cultured DRG neurons from young adult mice. This effect required Src family kinase (SFK) activity. In neurons from GFRalpha2-deficient mice, NRTN did not significantly promote axonal initiation. GDNF and NRTN induced extensive lamellipodia formation on neuronal somata and growth cones. This study suggested that GDNF, NRTN and ARTN may serve as stimulators of nerve regeneration under posttraumatic conditions. Consequently we studied the convergence of signaling pathways induced by NTFs and the ECM molecule laminin in the intracellular signaling network that regulates axonal growth. We demonstrated that co-stimulation of DRG neurons with NTFs (GDNF, NRTN or nerve growth factor (NGF)) and laminin leads to axonal growth that requires activation of SFKs. A different, SFK-independent signaling pathway evoked axonal growth on laminin in the absence of the NTFs. In contrast, axonal branching was regulated by SFKs both in the presence and in the absence of NGF. We proposed and experimentally verified a Boolean model of the signaling network triggered by NTFs and laminin. Our results put forward an approach for predictable, Boolean logics-driven pharmacological manipulation of a complex signaling network. Finally we found that N-syndecan, the receptor for the ECM component HB-GAM was required for the survival of neonatal sensory neurons in vitro. We demonstrated massive cell death of cultured DRG neurons from mice deficient in the N-syndecan gene as compared to wild type controls. Importantly, this cell death could not be prevented by NGF the neurotrophin which activates multiple anti-apoptotic cascades in DRG neurons. The survival deficit was observed during first postnatal week. By contrast, DRG neurons from young adult N-syndecan knock-out mice exhibited normal survival. This study identifies a completely new syndecan-dependent type of signaling that regulates cell death in neurons.

Novel matrix metalloproteinases in intestinal inflammation and in cancer of the gastrointestinal tract

Matrix metalloproteinases (MMPs) comprise a family of 23 zinc-dependent human endopeptidases that can degrade virtually all components of the extracellular matrix (ECM). They are classified into eight subgroups according to their structure and into six subgroups based on their substrate-specificity. MMPs have been implicated in inflammation, tissue destruction, cell migration, arthritis, vascular remodeling, angiogenesis, and tumor growth and invasion. MMPs are inhibited by their natural inhibitors, tissue inhibitors of metalloproteinases (TIMPs). Different MMPs function in the same tasks depending on the tissue or cancer subtype. I investigated the role of recently discovered MMPs, especially MMPs-19 and -26, in intestinal inflammation, in intestinal and cutaneous wound healing, and in intestinal cancer. Several MMPs and TIMPs were studied to determine their exact location at tissue level and to obtain information on possible functions of MMPs in such tissues and diseases as the healthy intestine, inflammatory bowel disease (IBD), neonatal necrotizing enterocolitis (NEC), pyoderma gangrenosum (PG), and colorectal as well as pancreatic cancers. In latent celiac disease (CD), I attempted to identify markers to predict later onset of CD in children and adolescents. The main methods used were immunohistochemistry, in situ hybridization, and Taqman RT-PCR. My results show that MMP-26 is important for re-epithelialization in intestinal and cutaneous wound healing. In colon and pancreatic cancers, MMP-26 seems to be a marker of invasive potential, although it is not itself expressed at the invasive front. MMP-21 is upregulated in pancreatic cancer and may be associated with tumor differentiation. MMPs-19 and -28 are associated with normal tissue turnover in the intestine, but they disappear in tumor progression as if they were protective markers . MMP-12 is an essential protease in intestinal inflammation and tissue destruction, as seen here in NEC and in previous CD studies. In patients with type 1 diabetes (T1D), MMPs-1, -3, and -12 were upregulated in the intestinal mucosa. Furthermore, MMP-7 was strongly elevated in NEC. In a model of aberrant wound repair, PG, MMPs-8, -9, and 10 and TNFα may promote ECM destruction, while absence of MMP-1 and MMP-26 from keratinocytes retards re-epithelialization. Based on my results, I suggest MMP-26 to be considered a putative marker for poor prognosis in pancreatic and colon cancer. However, since it functions differently in various tissues and tumor subtypes, this use cannot be generalized. Furthermore, MMP-26 is a beneficial marker for wound healing if expressed by migrating epithelial cells. MMP-12 expression in latent CD patients warrants research in a larger patient population to confirm its role as a specific marker for CD in pathologically indistinct cases. MMP-7 should be considered one of the most crucial proteases in NEC-associated tissue destruction; hence, specific inhibitors of this MMP are worth investigating. In PG, TNFα inhibitors are potential therapeutic agents, as shown already in clinical trials. In conclusion, studies of several MMPs in specific diseases and in healthy tissues are needed to elucidate their roles at the tissue level. MMPs and TIMPs are not exclusively destructive or reparative in tissues. They seem to function differently in different tissues. To identify selective MMP inhibitors, we must thoroughly understand the MMP profile (degradome) and their functions in various organs not to interfere with normal reparative functions during wound repair or beneficial host-response effects during cancer initiation and growth.

Matrix Metalloproteinases and their Tissue Inhibitors as Biomarkers in Ulcerative Colitis and Crohn s Disease

Matrix metalloproteinases (MMPs) represent a family of 23 metalloendopeptidases, collectively capable of degrading all components of the extracellular matrix. MMPs have been implicated in several inflammatory processes such as arthritis, atherosclerosis, and even carcinomas. They are also involved in several beneficial activities such as epithelial repair. MMPs are inhibited by endogenous tissue inhibitors of matrix metalloproteinases (TIMP). In this study, MMPs were investigated in intestinal mucosa of inflammatory bowel diseases (IBD), chronic intestinal disorders. The main focus was to characterize mucosal inflammation in the intestine, but also cutaneous pyoderma gangrenosum (PG), to assess similarites with IBD inflammation. MMPs and TIMPs were mainly examined in colonic mucosa, in adult Crohn s disease (CD), and paediatric CD, ulcerative colitis (UC), and indeterminate colitis (IC). Ileal pouch mucosa of proctocolectomized paediatric onset IBD patients was also investigated to characterize pouch mucosa. The focus was on finding specific MMPs that could act as markers to differentiate between different IBD disorders, and MMPs that could be implied as markers for tissue injury, potentially serving as targets for MMP-inhibitors. All examinations were performed using immunohistochemistry. The results show that immunosuppressive agents decrease stromal expression of MMP-9 and -26 that could serve as specific targets for MMP-inhibitors in treating CD. In paediatric colonic inflammation, MMP-10 and TIMP-3 present as molecular markers for IBD inflammation, and MMP-7 for CD. MMP expression in the the pouch mucosa could not be classified as strictly IBD- or non-IBD-like. For the first time, this study describes the expression of MMP-3, -7, -9, -12, and TIMP-2 and -3 in pouch mucosa. The MMP profile in PG bears resemblance to both intestinal IBD inflammation and cutaneous inflammation. Based on the results, MMPs and their inhibitors emerge as promising tools in the differential diagnosis of IBD and characterization of the disease subtype, although further research is necessary. Furthermore, the expression of several MMPs in pouch has been described for the first time. While further research is warranted, the findings contribute to a better understanding of events occurring in IBD mucosa, as well as pyoderma gangrenosum.

Matrix metalloproteinases as biomarkers in premalignant and malignant tumors of the human skin

The incidence of non-melanoma skin cancer is increasing worldwide. Basal cell carcinoma followed by squamous cell carcinoma and malignant melanoma are the most frequent skin tumors. Immunosuppressed patients have an increased risk of neoplasia, of which non-melanoma skin cancer is the most common. Matrix metalloproteinases (MMPs) are proteolytic enzymes that collectively are capable of degrading virtually all components of the extracellular matrix. MMPs can also process substrates distinct from extracellular matrix proteins and influence cell proliferation, differentiation, angiogenesis, and apoptosis. MMP activity is regulated by their natural inhibitors, tissue inhibitors of metallopro-teinases (TIMPs). In this study, the expression patterns of MMPs, TIMPs, and certain cancer-related molecules were investigated in premalignant and malignant lesions of the human skin. As methods were used immunohistochemisty, in situ hybridization, and reverse transcriptase polymerase chain reaction (RT-PCR) from the cell cultures. Our aim was to evaluate the expression pattern of MMPs in extramammary Paget's disease in order to find markers for more advanced tumors, as well as to shed light on the origin of this rare neoplasm. Novel MMPs -21, -26, and -28 were studied in melanoma cell culture, in primary cutaneous melanomas, and their sentinel nodes. The MMP expression profile in keratoacanthomas and well-differentiated squamous cell carcinomas was analyzed to find markers to differentiate benign keratinocyte hyperproliferation from malignantly transformed cells. Squamous cell carcinomas of immunosuppressed organ transplant recipients were compared to squamous cell carcinomas of matched immunocompetent controls to investigate the factors explaining their more aggressive nature. We found that MMP-7 and -19 proteins are abundant in extramammary Paget's disease and that their presence may predict an underlying adenocarcinoma in these patients. In melanomas, MMP-21 was upregulated in early phases of melanoma progression, but disappeared from the more aggressive tumors with lymph node metastases. The presence of MMP-13 in primary melanomas and lymph node metastases may relate to more aggressive disease. In keratoacanthomas, the expression of MMP-7 and -9 is rare and therefore should raise a suspicion of well-differentiated squamous cell carcinomas. Furthermore, MMP-19 and p16 were observed in benign keratinocyte hyperproliferation of keratoacanthomas, whereas they were generally lost from malignant keratinocytes of SCCs. MMP-26 staining was significantly stronger in squamous cell carcinomas and Bowen s disease samples of organ transplant recipients and it may contribute to the more aggressive nature of squamous cell carcinomas in immunosuppressed patients. In addition, the staining for MMP-9 was significantly stronger in macrophages surrounding the tumors of the immunocompetent group and in neutrophils of those patients on cyclosporin medication. In conclusion, based on our studies, MMP-7 and -19 might serve as biomarkers for more aggressive extramammary Paget's disease and MMP-21 for malignant transformation of melanocytes. MMP -7, -9, and -26, however, could play an important role in the pathobiology of keratinocyte derived malignancies.

Preterm delivery and selected biomarkers : phosphorylated insulin-like growth factor-binding protein-1 and matrix metalloproteinase-8 – in cervical fluid

Premature delivery is a major cause of neonatal morbidity and mortality. The incidence of premature deliveries has increased around the world. In Finland 5.3%, or about 3,000 children per year are born prematurely, before 37 weeks of gestation. The corresponding figure in the United States is about 13%. The morbidity and mortality are highest among infants delivered before 32 weeks of gestation - about 600 children each year in Finland. Approximately 70% of premature deliveries are unexplained. Preterm delivery can be caused by an asympto-matic infection between uterus and the fetal membranes, such can begin already in early pregnancy. It is difficult to predict preterm delivery, and many patients are therefore unnecessarily admitted to hospital for observation and exposed to medical treatments. On the other hand, the high risk women should be identified early for the best treatment of the mother and preterm infant. --- In the prospective study conducted at the Department of Obstetric and Gynecology, Helsinki University Central Hospital two biochemical inflammation related markers were measured in the lower genital tract fluids of asymp-tomatic women in early and mid pregnancy in an order to see whether these markers could identify women with an increased risk of preterm delivery. These biomarkers were phosphorylated insulin-like growth factor binding protein-1 (phIGFBP-1) and matrix metalloproteinase-8 (MMP-8). The study involved 5180 asymptomatic pregnant women, examined during the first and second ultrasound screening visits. The study samples were taken from the vagina and cervicix. In addition, 246 symptomatic women were studied (pregnancy weeks 22 – 34). The study showed that increased phIGFBP-1 concentration in cervical canal fluid in early pregnancy increased the risk for preterm delivery. The risk for very premature birth (before 32 weeks of gestation) was nearly four-fold. Low MMP-8 concentration in mid pregnancy increased the risk of subsequent premature preterm rupture of fetal membranes (PPROM). Significantly high MMP-8 concentrations in the cervical fluid increased the risk for prema-ture delivery initiated by preterm labour with intact membranes. Among women with preterm contractions the shortened cervical length measured by ultrasound and elevated cervical fluid phIGFBP-1 both predicted premature delivery. In summary, because of the relatively low sensitivity of cervical fluid phIGFBP-1 this biomarker is not suitable for routine screening, but provides an additional tool in assessing the risk of preterm delivery. Cervical fluid MMP-8 is not useful in early or mid pregnancy in predicting premature delivery because of its dual role. Further studies on the role of MMP-8 are therefore needed. Our study confirms that phIGFBP-1 testing is useful in predicting pre-term delivery.