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.

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

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Composition operators on vector-valued BMOA and related function spaces

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

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.

Convex-transitive Banach spaces and their hyperplanes

The topic of this dissertation is the geometric and isometric theory of Banach spaces. This work is motivated by the known Banach-Mazur rotation problem, which asks whether each transitive separable Banach space is isometrically a Hilbert space. A Banach space X is said to be transitive if the isometry group of X acts transitively on the unit sphere of X. In fact, some weaker symmetry conditions than transitivity are studied in the dissertation. One such condition is an almost isometric version of transitivity. Another investigated condition is convex-transitivity, which requires that the closed convex hull of the orbit of any point of the unit sphere under the rotation group is the whole unit ball. Following the tradition developed around the rotation problem, some contemporary problems are studied. Namely, we attempt to characterize Hilbert spaces by using convex-transitivity together with the existence of a 1-dimensional bicontractive projection on the space, and some mild geometric assumptions. The convex-transitivity of some vector-valued function spaces is studied as well. The thesis also touches convex-transitivity of Banach lattices and resembling geometric cases.

Aspects of atomic decompositions and Bergman projections

The concept of an atomic decomposition was introduced by Coifman and Rochberg (1980) for weighted Bergman spaces on the unit disk. By the Riemann mapping theorem, functions in every simply connected domain in the complex plane have an atomic decomposition. However, a decomposition resulting from a conformal mapping of the unit disk tends to be very implicit and often lacks a clear connection to the geometry of the domain that it has been mapped into. The lattice of points, where the atoms of the decomposition are evaluated, usually follows the geometry of the original domain, but after mapping the domain into another this connection is easily lost and the layout of points becomes seemingly random. In the first article we construct an atomic decomposition directly on a weighted Bergman space on a class of regulated, simply connected domains. The construction uses the geometric properties of the regulated domain, but does not explicitly involve any conformal Riemann map from the unit disk. It is known that the Bergman projection is not bounded on the space L-infinity of bounded measurable functions. Taskinen (2004) introduced the locally convex spaces LV-infinity consisting of measurable and HV-infinity of analytic functions on the unit disk with the latter being a closed subspace of the former. They have the property that the Bergman projection is continuous from LV-infinity onto HV-infinity and, in some sense, the space HV-infinity is the smallest possible substitute to the space H-infinity of analytic functions. In the second article we extend the above result to a smoothly bounded strictly pseudoconvex domain. Here the related reproducing kernels are usually not known explicitly, and thus the proof of continuity of the Bergman projection is based on generalised Forelli-Rudin estimates instead of integral representations. The minimality of the space LV-infinity is shown by using peaking functions first constructed by Bell (1981). Taskinen (2003) showed that on the unit disk the space HV-infinity admits an atomic decomposition. This result is generalised in the third article by constructing an atomic decomposition for the space HV-infinity on a smoothly bounded strictly pseudoconvex domain. In this case every function can be presented as a linear combination of atoms such that the coefficient sequence belongs to a suitable Köthe co-echelon space.

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.

Surface Proteins of Lactobacillus crispatus: Adhesive Properties and Cell Wall Anchoring

Bacterial surface-associated proteins are important in communication with the environment and bacteria-host interactions. In this thesis work, surface molecules of Lactobacillus crispatus important in host interaction were studied. The L. crispatus strains of the study were known from previous studies to be efficient in adhesion to intestinal tract and ECM. L. crispatus JCM 5810 possess an adhesive surface layer (S-layer) protein, whose functions and domain structure was characterized. We cloned two S-layer protein genes (cbsA; collagen-binding S-layer protein A and silent cbsB) and identified the protein region in CbsA important for adhesion to host tissues, for polymerization into a periodic layer as well as for attachment to the bacterial cell surface. The analysis was done by extensive mutation analysis and by testing His6-tagged fusion proteins from recombinant Escherichia coli as well as by expressing truncated CbsA peptides on the surface of Lactobacillus casei. The N-terminal region (31-274) of CbsA showed efficient and specific binding to collagens, laminin and extracellular matrix on tissue sections of chicken intestine. The N-terminal region also contained the information for formation of periodic S-layer polymer. This region is bordered at both ends by a conserved short region rich in valines, whose substitution to leucines drastically affected the periodic polymer structure. The mutated CbsA proteins that failed to form a periodic polymer, did not bind collagens, which indicates that the polymerized structure of CbsA is needed for collagen-binding ability. The C-terminal region, which is highly identical in S-layer proteins of L. crispatus, Lactobacillus acidophilus and Lactobacillus helveticus, was shown to anchor the protein to the bacterial cell wall. The C-terminal CbsA peptide specifically bound to bacterial teichoic acid and lipoteichoic acids. In conclusion, the N-terminal domain of the S-layer protein of L. crispatus is important for polymerization and adhesion to host tissues, whereas the C-terminal domain anchors the protein to bacterial cell-wall teichoic acids. Lactobacilli are fermentative organisms that effectively lower the surrounding pH. While this study was in progress, plasminogen-binding proteins enolase and glyceraldehyde-3-phosphate dehydrogenase (GAPDH) were identified in the extracellular proteome of L. crispatus ST1. In this work, the cell-wall association of enolase and GAPDH were shown to rely on pH-reversible binding to the cell-wall lipoteichoic acids. Enolase from L. crispatus was functionally compared with enolase from L. johnsonii as well as from pathogenic streptococci (Streptococcus pneumoniae, Streptococcus pyogenes) and Staphylococcus aureus. His6-enolases from commensal lactobacilli bound human plasminogen and enhanced its activation by human plasminogen activators similarly to, or even better than, the enolases from pathogens. Similarly, the His6-enolases from lactobacilli exhibited adhesive characteristics previously assigned to pathogens. The results call for more detailed analyses of the role of the host plasminogen system in bacterial pathogenesis and commensalism as well of the biological role and potential health risk of the extracellular proteome in lactobacilli.

Measurement of the top quark mass at CDF using the “neutrino ϕ weighting” template method on a lepton plus isolated track sample

We present a measurement of the top quark mass with t-tbar dilepton events produced in p-pbar collisions at the Fermilab Tevatron $\sqrt{s}$=1.96 TeV and collected by the CDF II detector. A sample of 328 events with a charged electron or muon and an isolated track, corresponding to an integrated luminosity of 2.9 fb$^{-1}$, are selected as t-tbar candidates. To account for the unconstrained event kinematics, we scan over the phase space of the azimuthal angles ($\phi_{\nu_1},\phi_{\nu_2}$) of neutrinos and reconstruct the top quark mass for each $\phi_{\nu_1},\phi_{\nu_2}$ pair by minimizing a $\chi^2$ function in the t-tbar dilepton hypothesis. We assign $\chi^2$-dependent weights to the solutions in order to build a preferred mass for each event. Preferred mass distributions (templates) are built from simulated t-tbar and background events, and parameterized in order to provide continuous probability density functions. A likelihood fit to the mass distribution in data as a weighted sum of signal and background probability density functions gives a top quark mass of $165.5^{+{3.4}}_{-{3.3}}$(stat.)$\pm 3.1$(syst.) GeV/$c^2$.