Playing Games on Sets and Models

The most prominent objective of the thesis is the development of the generalized descriptive set theory, as we call it. There, we study the space of all functions from a fixed uncountable cardinal to itself, or to a finite set of size two. These correspond to generalized notions of the universal Baire space (functions from natural numbers to themselves with the product topology) and the Cantor space (functions from natural numbers to the {0,1}-set) respectively. We generalize the notion of Borel sets in three different ways and study the corresponding Borel structures with the aims of generalizing classical theorems of descriptive set theory or providing counter examples. In particular we are interested in equivalence relations on these spaces and their Borel reducibility to each other. The last chapter shows, using game-theoretic techniques, that the order of Borel equivalence relations under Borel reduciblity has very high complexity. The techniques in the above described set theoretical side of the thesis include forcing, general topological notions such as meager sets and combinatorial games of infinite length. By coding uncountable models to functions, we are able to apply the understanding of the generalized descriptive set theory to the model theory of uncountable models. The links between the theorems of model theory (including Shelah's classification theory) and the theorems in pure set theory are provided using game theoretic techniques from Ehrenfeucht-Fraïssé games in model theory to cub-games in set theory. The bottom line of the research declairs that the descriptive (set theoretic) complexity of an isomorphism relation of a first-order definable model class goes in synch with the stability theoretical complexity of the corresponding first-order theory. The first chapter of the thesis has slightly different focus and is purely concerned with a certain modification of the well known Ehrenfeucht-Fraïssé games. There we (me and my supervisor Tapani Hyttinen) answer some natural questions about that game mainly concerning determinacy and its relation to the standard EF-game

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.

Positional cloning and pathway analysis of the asthma susceptibility gene, NPSR1

In the present study, we identified a novel asthma susceptibility gene, NPSR1 (neuropeptide S receptor 1) on chromosome 7p14.3 by the positional cloning strategy. An earlier significant linkage mapping result among Finnish Kainuu asthma families was confirmed in two independent cohorts: in asthma families from Quebec, Canada and in allergy families from North Karelia, Finland. The linkage region was narrowed down to a 133-kb segment by a hierarchial genotyping method. The observed 77-kb haplotype block showed 7 haplotypes and a similar risk and nonrisk pattern in all three populations studied. All seven haplotypes occur in all three populations at frequences > 2%. Significant elevated relative risks were detected for elevated total IgE (immunoglobulin E) or asthma. Risk effects of the gene variants varied from 1.4 to 2.5. NPSR1 belongs to the G protein-coupled receptor (GPCR) family with a topology of seven transmembrane domains. NPSR1 has 9 exons, with the two main transcripts, A and B, encoding proteins of 371 and 377 amino acids, respectively. We detected a low but ubiquitous expression level of NPSR1-B in various tissues and endogenous cell lines while NPSR1-A has a more restricted expression pattern. Both isoforms were expressed in the lung epithelium. We observed aberrant expression levels of NPSR1-B in smooth muscle in asthmatic bronchi as compared to healthy. In an experimental mouse model, the induced lung inflammation resulted in elevated Npsr1 levels. Furthermore, we demonstrated that the activation of NPSR1 with its endogenous agonist, neuropeptide S (NPS), resulted in a significant inhibition of the growth of NPSR1-A overexpressing stable cell lines (NPSR1-A cells). To determine which target genes were regulated by the NPS-NPSR1 pathway, NPSR1-A cells were stimulated with NPS, and differentially expressed genes were identified using the Affymetrix HGU133Plus2 GeneChip. A total of 104 genes were found significantly up-regulated and 42 down-regulated 6 h after NPS administration. The up-regulated genes included many neuronal genes and some putative susceptibility genes for respiratory disorders. By Gene Ontology enrichment analysis, the biological process terms, cell proliferation, morphogenesis and immune response were among the most altered. The expression of four up-regulated genes, matrix metallopeptidase 10 (MMP10), INHBA (activin A), interleukin 8 (IL8) and EPH receptor A2 (EPHA2), were verified and confirmed by quantitative reverse-transcriptase-PCR. In conclusion, we identified a novel asthma susceptibility gene, NPSR1, on chromosome 7p14.3. NPS-NPSR1 represents a novel pathway that regulates cell proliferation and immune responses, and thus may have functional relevance in the pathogenesis of asthma.

Search for single top quark production in pp̅ collisions at √s=1.96 TeV in the missing transverse energy plus jets topology

We report a search for single top quark production with the CDF II detector using 2.1 fb-1 of integrated luminosity of pbar p collisions at sqrt{s}=1.96 TeV. The data selected consist of events characterized by large energy imbalance in the transverse plane and hadronic jets, and no identified electrons and muons, so the sample is enriched in W -> tau nu decays. In order to suppress backgrounds, additional kinematic and topological requirements are imposed through a neural network, and at least one of the jets must be identified as a b-quark jet. We measure an excess of signal-like events in agreement with the standard model prediction, but inconsistent with a model without single top quark production by 2.1 standard deviations (sigma), with a median expected sensitivity of 1.4 sigma. Assuming a top quark mass of 175 GeV/c2 and ascribing the excess to single top quark production, the cross section is measured to be 4.9+2.5-2.2(stat+syst)pb, consistent with measurements performed in independent datasets and with the standard model prediction.

Observation of single top quark production and measurement of |Vtb| with CDF

We report the observation of electroweak single top quark production in 3.2  fb-1 of pp̅ collision data collected by the Collider Detector at Fermilab at √s=1.96  TeV. Candidate events in the W+jets topology with a leptonically decaying W boson are classified as signal-like by four parallel analyses based on likelihood functions, matrix elements, neural networks, and boosted decision trees. These results are combined using a super discriminant analysis based on genetically evolved neural networks in order to improve the sensitivity. This combined result is further combined with that of a search for a single top quark signal in an orthogonal sample of events with missing transverse energy plus jets and no charged lepton. We observe a signal consistent with the standard model prediction but inconsistent with the background-only model by 5.0 standard deviations, with a median expected sensitivity in excess of 5.9 standard deviations. We measure a production cross section of 2.3-0.5+0.6(stat+sys)  pb, extract the value of the Cabibbo-Kobayashi-Maskawa matrix element |Vtb|=0.91-0.11+0.11(stat+sys)±0.07  (theory), and set a lower limit |Vtb|>0.71 at the 95% C.L., assuming mt=175  GeV/c2.

Observation of single top quark production and measurement of |Vtb| with CDF

We report the observation of electroweak single top quark production in 3.2 fb-1 of ppbar collision data collected by the Collider Detector at Fermilab at sqrt{s}=1.96 TeV. Candidate events in the W+jets topology with a leptonically decaying W boson are classified as signal-like by four parallel analyses based on likelihood functions, matrix elements, neural networks, and boosted decision trees. These results are combined using a super discriminant analysis based on genetically evolved neural networks in order to improve the sensitivity. This combined result is further combined with that of a search for a single top quark signal in an orthogonal sample of events with missing transverse energy plus jets and no charged lepton. We observe a signal consistent with the standard model prediction but inconsistent with the background-only model by 5.0 standard deviations, with a median expected sensitivity in excess of 5.9 standard deviations. We measure a production cross section of 2.3+0.6-0.5(stat+sys) pb, extract the CKM matrix element value |Vtb|=0.91+0.11-0.11 (stat+sys)+-0.07(theory), and set a lower limit |Vtb|>0.71 at the 95% confidence level, assuming m_t=175 GeVc^2.