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This self-contained encyclopedic monograph gives a detailed introduction to Bézout equations and stable ranks, encompassing and explaining needed topological, analytical, and algebraic tools and methods. Some of the highlights included are Carleson's corona theorem and the Bass, topological, and matricial stable ranks. The first volume focusses on topological structures, Banach algebras, and advanced function theory, thus preparing the stage for the algebraic structures in the second volume towards examining stable ranks with analytic methods. The main emphasis is laid on algebras of holomorphic functions. Often a new approach is presented or at least a different angle of sight, which makes the book attractive both for researchers and students interested in these active fields of research.

Functions of complex variables. --- Functional analysis. --- Functions of a Complex Variable. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Complex variables --- Elliptic functions --- Functions of real variables --- Funcions de variables complexes --- Anàlisi funcional --- Càlcul funcional --- Càlcul de variacions --- Àlgebres de Hilbert --- Àlgebres topològiques --- Anàlisi funcional no lineal --- Anàlisi microlocal --- Espais analítics --- Espais de Hardy --- Espais d'Orlicz --- Espais funcionals --- Espais vectorials normats --- Espais vectorials --- Filtres digitals (Matemàtica) --- Funcionals --- Funcions vectorials --- Multiplicadors (Anàlisi matemàtica) --- Pertorbació (Matemàtica) --- Teoria d'operadors --- Teoria de distribucions (Anàlisi funcional) --- Teoria de functors --- Teoria de l'aproximació --- Teoria del funcional de densitat --- Teoria espectral (Matemàtica) --- Equacions funcionals --- Equacions integrals --- Anàlisi complexa --- Funció d'una variable complexa --- Variables complexes --- Aplicacions quasiconformes --- Funcions abelianes --- Funcions analítiques --- Funcions convexes --- Funcions de diverses variables complexes --- Funcions enteres --- Funcions meromorfes --- Funcions univalents --- Grups discontinus --- Invariants conformes --- Problemes de contorn --- Sèries de Lie --- Teoria geomètrica de funcions --- Funcions de variables reals --- Funcions el·líptiques --- Sistemes dinàmics complexos

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This monograph offers the first systematic treatment of the theory of minimal surfaces in Euclidean spaces by complex analytic methods, many of which have been developed in recent decades as part of the theory of Oka manifolds (the h-principle in complex analysis). It places particular emphasis on the study of the global theory of minimal surfaces with a given complex structure. Advanced methods of holomorphic approximation, interpolation, and homotopy classification of manifold-valued maps, along with elements of convex integration theory, are implemented for the first time in the theory of minimal surfaces. The text also presents newly developed methods for constructing minimal surfaces in minimally convex domains of Rn, based on the Riemann–Hilbert boundary value problem adapted to minimal surfaces and holomorphic null curves. These methods also provide major advances in the classical Calabi–Yau problem, yielding in particular minimal surfaces with the conformal structure of any given bordered Riemann surface. Offering new directions in the field and several challenging open problems, the primary audience of the book are researchers (including postdocs and PhD students) in differential geometry and complex analysis. Although not primarily intended as a textbook, two introductory chapters surveying background material and the classical theory of minimal surfaces also make it suitable for preparing Masters or PhD level courses.

Global analysis (Mathematics). --- Manifolds (Mathematics). --- Functions of complex variables. --- Global Analysis and Analysis on Manifolds. --- Several Complex Variables and Analytic Spaces. --- Complex variables --- Elliptic functions --- Functions of real variables --- Geometry, Differential --- Topology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Global analysis (Mathematics) --- Manifolds (Mathematics) --- Minimal surfaces. --- Surfaces, Minimal --- Maxima and minima --- Anàlisi global (Matemàtica) --- Funcions de variables complexes --- Varietats (Matemàtica) --- Superfícies mínimes --- Màxims i mínims --- Aplicacions de Gauss --- Anàlisi complexa --- Funció d'una variable complexa --- Variables complexes --- Aplicacions quasiconformes --- Funcions abelianes --- Funcions analítiques --- Funcions convexes --- Funcions de diverses variables complexes --- Funcions enteres --- Funcions meromorfes --- Funcions univalents --- Grups discontinus --- Invariants conformes --- Problemes de contorn --- Sèries de Lie --- Teoria geomètrica de funcions --- Funcions de variables reals --- Funcions el·líptiques --- Sistemes dinàmics complexos --- Topologia diferencial --- Geodèsiques (Matemàtica) --- Geometria espectral --- Teoria del punt crític (Anàlisi matemàtica) --- Varietats de dimensió infinita --- Varietats analítiques --- Geometria diferencial --- Topologia --- Varietats topològiques --- Catàstrofes (Matemàtica) --- Homeomorfismes --- Subvarietats (Matemàtica) --- Topologia de baixa dimensió --- Tor (Geometria) --- Varietats complexes --- Varietats de Calabi-Yau --- Varietats de Grassmann --- Varietats diferenciables --- Varietats de Kähler --- Varietats simplèctiques