The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df(0,t) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Carathéodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh(0,t)=A where inf {Re<Az,z>: ||z| =1}> 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Carathéodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Carathéodory mappings on the ball starting from a known example of extreme Carathéodory function in higher dimensions.

The first part of the thesis deals with aspects of Loewner theory in several complex variables. First we show that a Loewner chain with minimal regularity assumptions (Df(0,t) of local bounded variation) satisfies an associated Loewner equation. Next we give a way of renormalizing a general Loewner chain so that it corresponds to the same increasing family of domains. To do this we will prove a generalization of the converse of Carathéodory's kernel convergence theorem. Next we address the problem of finding a Loewner chain solution to a given Loewner chain equation. The main result is a complete solution in the case when the infinitesimal generator satisfies Dh(0,t)=A where inf {Re<Az,z>: ||z| =1}> 0. We will see that the existence of a bounded solution depends on the real resonances of A, but there always exists a polynomially bounded solution. Finally we discuss some properties of classes of biholomorphic mappings associated to A-normalized Loewner chains. In particular we give a characterization of the compactness of the class of spirallike mappings in terms of the resonance of A. The second part of the thesis deals with the problem of finding examples of extreme points for some classes of mappings. We see that straightforward generalizations of one dimensional extreme functions give examples of extreme Carathéodory mappings and extreme starlike mappings on the polydisc, but not on the ball. We also find examples of extreme Carathéodory mappings on the ball starting from a known example of extreme Carathéodory function in higher dimensions.

L’objectif de cette thèse est d’étudier divers concepts de positivité en géométrie kählerienne. En particulier,pour une variété kählerienne compacte de dimension n, nous étudions la positivité des classes transcendantes de type (1,1) et (n-1, n-1) - ces classes comprennent donc en particulier les classesde diviseurs et les classes de courbes. / The goal of this thesis is to study various positivity concepts in Kähler geometry. In particular, for a compact Kähler manifold of dimension n, we study the positivity of transcendental (1,1) and (n-1, n-1) classes. These objects include the divisor classes and curve classes over smooth complex projective varieties.

We give a proof of the Oka-Weil theorem which states that on compact, polynomially convex subsets of Cn, holomorphic functions can be approximated uniformly by holomorphic polynomials. / Vi ger ett bevis av Oka-Weil sats som säger att på kompakta och polynomkonvexa delmängder av Cn kan holomorfa funktioner approximeras likformigt med holomorfa polynom.

In this thesis we have investigated two different types of problems in multivariable operator theory. The first one deals with the defect sequence for contractive tuples and maximal con-tractive tuples. These condone deals with the wandering subspaces of the Bergman space and the Dirichlet space over the polydisc. These are described in thefollowing two sections. (I) The Defect Sequence for ContractiveTuples LetT=(T1,...,Td)bead-tuple of bounded linear operators on some Hilbert space H. We say that T is a row contraction, or, acontractive tuplei f the row operator (Pl refer the abstract pdf file)

Let <i>φ</i> be a self-map of <b>B</b>₂, the unit ball in <b>C</b>². We investigate the equation <i>C_φf</i>=<i>λf</i> where we define <i>C_φf </i>: -<i>f◦φ</i>, with <i>f a</i> function in the Drury Arves on Space. After imposing conditions to keep <i>C_φ</i> bounded and well-behaved, we solve the equation <i>C_φf - λf </i>and determine the spectrum <i>σ</i>(<i>C_φ</i>) in the case where there is no interior fixed point and boundary fixed point without multiplicity. We then investigate the existence of one-parameter semigroups for such maps and discuss some generalizations.

Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.

La première partie présente des calculs explicites de terminaison effective de l'algorithme de Kohn proposée par Siu. Dans la deuxième partie, nous étudions la géométrie des hypersurfaces réelles dans Cⁿ, et nous calculons des invariants explicites avec la méthode d'équivalences de Cartan pour déterminer les lieux CR-ombilics. / The first part of the thesis consists of calculations around Siu's effective termination of Kohn's algorithm. The second part of the thesis studies the CR real hypersurfaces in complex spaces and calculates various explicit invariants using Cartan's equivalence method to study CR-umbilical points.

Indiana University-Purdue University Indianapolis (IUPUI) / The objective of this thesis is to compare automated methods for performing analysis of flow cytometry data. Flow cytometry is an important and efficient tool for analyzing the characteristics of cells. It is used in several fields, including immunology, pathology, marine biology, and molecular biology. Flow cytometry measures light scatter from cells and fluorescent emission from dyes which are attached to cells. There are two main tasks that must be performed. The first is the adjustment of measured fluorescence from the cells to correct for the overlap of the spectra of the fluorescent markers used to characterize a cell’s chemical characteristics. The second is to use the amount of markers present in each cell to identify its phenotype. Several methods are compared to perform these tasks. The Unconstrained Least Squares, Orthogonal Subspace Projection, Fully Constrained Least Squares and Fully Constrained One Norm methods are used to perform compensation and compared. The fully constrained least squares method of compensation gives the overall best results in terms of accuracy and running time. Spectral Clustering, Gaussian Mixture Modeling, Naive Bayes classification, Support Vector Machine and Expectation Maximization using a gaussian mixture model are used to classify cells based on the amounts of dyes present in each cell. The generative models created by the Naive Bayes and Gaussian mixture modeling methods performed classification of cells most accurately. These supervised methods may be the most useful when online classification is necessary, such as in cell sorting applications of flow cytometers. Unsupervised methods may be used to completely replace manual analysis when no training data is given. Expectation Maximization combined with a cluster merging post-processing step gives the best results of the unsupervised methods considered.