Complex analysis, complex variables Books

Book SynopsisThis book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications.As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four.This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general.Table of ContentsIntroduction.- 1.Definitions and Basic Results.- 2.Regular Power Series.- 3.Zeros.- 4.Infinite Products.- 5.Singularities.- 6.Integral Representations.- 7.Maximum Modulus Theorem and Applications.- 8.Spherical Series and Differential.- 9.Fractional Transformations and the Unit Ball.- 10.Generalizations.- 11. Function Theory over Non-symmetric Slice Domains.-12. Applications.- Bibliography.- Index.

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Book SynopsisMapping Properties of Maximal Functions on Graded Lie Groups.- An Invitation to Quantum Field Theory and to its Interplay with Microlocal Analysis and PDEs.- Fourier Analysis via Mild Distributions: Group Theoretical Aspects.- On the Theory of Functions of Omega-Bounded Type.- Subsequences of Sequences of Multiple Partial Trigonometric Fourier Sums.- A Vectorial Free Boundary Transmission Problem (A Short Exposition).- On the ??2-Analogue of the Inverse Source Heat Equation.- On Typical and Atypical Asymptotic Behavior of Singular Solutions to Emden–Fowler Type Equations.- Some Harmonic Bergman-Type Projections on Besov and Bloch Spaces.- On a Dirichlet Problem for a Properly Elliptic Equation in the Space of Continuous Functions, in the Case of Multiple Roots.- On the Behavior of Fourier Coefficients in the Trigonometric System.- Discrete-Time Replicator Equations, Gradient Vector Fields of Nonlinear Mappings, and Optimal Transport Networks.- Deviation Identity for Linear Differential Operators and Its Application to Obstacle Problems.- Nonlinear Approximation with Respect to the Walsh Generalized System.- On the Convergence of Negative-Order Cesaro Means of Fourier and Fourier-Walsh Series.- Irreversibility of a Classical Three-Body Problem: Complexity of a Low-Dimensional System.- On Polynomial Solutions of a PDE with Constant Coefficients.- On Weighted Integral Operators for Solution of ??− Equation in the Siegel Domain of ??n.- On Directionally-Differentiable Selections of Set-Valued Mappings.- On the Convergence of Hard Sampling Operators.- Orientation-Dependent Section Distributions for Convex Bodies.- On the Issues of Modeling the Elimination of Deadlock Situations and Synchronization Problems Using Petri Nets.- Generalized Abel-Plana Formula as a Renormalization Tool in Quantum Field Theory.- On the Universal Functions for Weighted Spaces.- On the Convergence Fourier-Vilenkin Series.- On the Universal Functions for Weighted Spaces.- Construction of a Mathematical Model and Optimization of the Bending of a Beam.- Delaunay Triangulation in Numerical Solution of Two-Dimensional Boundary Value Problems.- Normal Solvability and Fredholm Properties for Regular Hypoelliptic Operators.

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Book SynopsisThe concept of moduli goes back to B. Riemann, who shows in [68] that the isomorphism class of a Riemann surface of genus 9 ~ 2 depends on 3g - 3 parameters, which he proposes to name "moduli". A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see [59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In­ variant Theory". We will recall the necessary tools from his book [59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in [78], a careful study of positivity proper­ ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf.Table of ContentsLeitfaden.- Classification Theory and Moduli Problems.- Notations and Conventions.- 1 Moduli Problems and Hilbert Schemes.- 1.1 Moduli Functors and Moduli Schemes.- 1.2 Moduli of Manifolds: The Main Results.- 1.3 Properties of Moduli Functors.- 1.4 Moduli Functors for ?-Gorenstein Schemes.- 1.5 A. Grothendieck’s Construction of Hilbert Schemes.- 1.6 Hilbert Schemes of Canonically Polarized Schemes.- 1.7 Hilbert Schemes of Polarized Schemes.- 2 Weakly Positive Sheaves and Vanishing Theorems.- 2.1 Coverings.- 2.2 Numerically Effective Sheaves.- 2.3 Weakly Positive Sheaves.- 2.4 Vanishing Theorems and Base Change.- 2.5 Examples of Weakly Positive Sheaves.- 3 D. Mumford’s Geometric Invariant Theory.- 3.1 Group Actions and Quotients.- 3.2 Linearizations.- 3.3 Stable Points.- 3.4 Properties of Stable Points.- 3.5 Quotients, without Stability Criteria.- 4 Stability and Ampleness Criteria.- 4.1 Compactifications and the Hilbert-Mumford Criterion.- 4.2 Weak Positivity of Line Bundles and Stability.- 4.3 Weak Positivity of Vector Bundles and Stability.- 4.4 Ampleness Criteria.- 5 Auxiliary Results on Locally Free Sheaves and Divisors.- 5.1 O. Gabber’s Extension Theorem.- 5.2 The Construction of Coverings.- 5.3 Singularities of Divisors.- 5.4 Singularities of Divisors in Flat Families.- 5.5 Vanishing Theorems and Base Change, Revisited.- 6 Weak Positivity of Direct Images of Sheaves.- 6.1 Variation of Hodge Structures.- 6.2 Weakly Semistable Reduction.- 6.3 Applications of the Extension Theorem.- 6.4 Powers of Dualizing Sheaves.- 6.5 Polarizations, Twisted by Powers of Dualizing Sheaves.- 7 Geometric Invariant Theory on Hilbert Schemes.- 7.1 Group Actions on Hilbert Schemes.- 7.2 Geometric Quotients and Moduli Schemes.- 7.3 Methods to Construct Quasi-Projective Moduli Schemes.- 7.4 Conditions for the Existence of Moduli Schemes: Case (CP).- 7.5 Conditions for the Existence of Moduli Schemes: Case (DP).- 7.6 Numerical Equivalence.- 8 Allowing Certain Singularities.- 8.1 Canonical and Log-Terminal Singularities.- 8.2 Singularities of Divisors.- 8.3 Deformations of Canonical and Log-Terminal Singularities.- 8.4 Base Change and Positivity.- 8.5 Moduli of Canonically Polarized Varieties.- 8.6 Moduli of Polarized Varieties.- 8.7 Towards Moduli of Canonically Polarized Schemes.- 9 Moduli as Algebraic Spaces.- 9.1 Algebraic Spaces.- 9.2 Quotients by Equivalence Relations.- 9.3 Quotients in the Category of Algebraic Spaces.- 9.4 Construction of Algebraic Moduli Spaces.- 9.5 Ample Line Bundles on Algebraic Moduli Spaces.- 9.6 Proper Algebraic Moduli Spaces for Curves and Surfaces.- References.- Glossary of Notations.

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Book SynopsisThis 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.Trade Review“This book is an excellent monograph on the theory of minimal surfaces in Euclidean spaces by using complex analytic methods. … The reviewer would recommend that not only experts in this field, but also graduate students and researchers in related fields read this book.” (Yu Kawakami, Mathematical Reviews, December, 2022)Table of Contents1 Fundamentals.- 2 Basics on Minimal Surfaces.- 3 Approximation and Interpolations Theorems for Minimal Surfaces.- 4 Complete Minimal Surfaces of Finite Total Curvature.- 5 The Gauss Map of a Minimal Surface.- 6 The Riemann–Hilbert Problem for Minimal Surfaces.- 7 The Calabi–Yau Problem for Minimal Surfaces.- 8 Minimal Surfaces in Minimally Convex Domains.- 9 Minimal Hulls, Null Hulls, and Currents.- References.- Index.

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Book SynopsisThe study of univalent functions dates back to the early years of the 20th century, and is one of the most popular research areas in complex analysis. This book is directed at introducing and bringing up to date current research in the area of univalent functions, with an emphasis on the important subclasses, thus providing an accessible resource suitable for both beginning and experienced researchers. Contents Univalent Functions – the Elementary Theory Definitions of Major Subclasses Fundamental Lemmas Starlike and Convex Functions Starlike and Convex Functions of Order α Strongly Starlike and Convex Functions Alpha-Convex Functions Gamma-Starlike Functions Close-to-Convex Functions Bazilevič Functions B1(α) Bazilevič Functions The Class U(λ) Convolutions Meromorphic Univalent Functions Loewner Theory Other Topics Open Problems

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Book SynopsisThis completely revised and updated edition of the one variable part of the author's classic older book "Iteration Theory of Holomorphic Maps on Taut Manifolds" presents the theory of holomorphic dynamical systems on hyperbolic Riemann surfaces from the very beginning of the subject up to the most recent developments. It is intended both as a reference book for the experts and as an accessible gateway to this beautiful theory for Master and Ph.D. students. It also contains extensive historical notes and references for further readings.

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Book SynopsisThis textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles.The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained.This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.Trade Review“The reorganization of the topics is fine surgical work. Several portions of the original monograph are sewn in a natural way in the new book, adding examples or additional text when necessary, and re-arranging the focus to make it a more friendly introduction to the subject. Careful attention to details and the required background makes the book under review accessible to an interested reader and could be a used as textbook for a course on abelian varieties.” (Felipe Zaldivar, MAA Reviews, June 18, 2023)Table of Contents1. Line Bundles on Complex Tori.- 2 Abelian Varieties.- 3 Moduli Spaces.- 4 Jacobian Varieties.- 5 Main Examples of Abelian Varieties.- 6 The Fourier Transform for Sheaves and Cycles.- 7 Introduction to the Hodge Conjecture for Abelian Varieties.

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Book SynopsisThis book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass.This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis.Trade Review“The Real and the Complex originated in a course Jeremy Gray gave to upper-level mathematics students and is part of a Springer series on undergraduate mathematics. An important aim of the book is to help the student understand how any given part of mathematics first arises and comes to fruition. … for those readers with some university-level background in mathematics who are interested in the technical contours of nineteenth-century science, Gray’s book will repay careful study and is highly recommended.” (Craig Fraser, Isis, Vol. 108 (2), June, 2017)“I view it as an extremely valuable addition to my library and a book that I am sure I will consult frequently in the future: this is a text that should serve well as a desk reference for a faculty member teaching either analysis (real or complex) or the history of mathematics. Certainly, no good university library should be without it.” (Mark Hunacek, MAA Reviews, maa.org, January, 2017)“Gray’s 29 chapters take us on quite a ride through the ideas, results, and difficulties of the major figures of nineteenth-century analysis. … Any mathematician who would like to know how analysis developed should find it interesting. Those who teach analysis can gain much insight from seeing the blind alleys as well as the great ideas … . The writing style is clear and unpretentious, as well as marked by immense learning. … I strongly recommend … .” (Judith V. Grabiner, Notices of the AMS, Vol. 64 (11), 2017)“The Real and the Complex … develops the subject of analysis by reviewing the works of some of the prominent mathematicians of the 19th century. … Summing Up: Recommended. Upper-division undergraduates, graduate students, researchers/faculty, professionals/practitioners.” (M. D. Sanford, Choice, Vol. 53 (9), May, 2016)“This is a history book on the development of mathematics in the 19th century. Each chapter is built up around one or a few mathematicians. … this book is not only an interesting read for the students who (have to) study it, but equally valuable for professional mathematicians.” (Adhemar Bultheel, European Mathematical Society, euro-math-soc.eu, February, 2016)Table of ContentsLagrange and foundations for the calculus.- Joseph Fourier.- Legendre.- Cauchy and continuity.- Cauchy: differentiation and integration.- Cauchy and complex functions to 1830.- Abel.- Jacobi.- Gauss.- Cauchy and complex function theory, 1830-1857.- Complex functions and elliptic integrals.- Revision.- Gauss, Green, and potential theory.- Dirichlet, potential theory, and Fourier series.- Riemann.- Riemann and complex function theory.- Riemann's later complex function theory.- Responses to Riemann's work.- Weierstrass.- Weierstrass's foundational results.- Revision { and assessment.- Uniform Convergence.- Integration and trigonometric series.- The fundamental theorem of the calculus.- The construction of the real numbers.- Implicit functions.- Towards Lebesgue's theory of integration.- Cantor, set theory, and foundations.- Topology.- Assessment.

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Book SynopsisAuthor received the 1962 Fields Medal Author received the 1988 Wolf Prize (honoring achievemnets of a lifetime) Author is leading expert in partial differential equationsTrade ReviewFrom the reviews: "...these volumes are excellently written and make for greatly profitable reading. For years to come they will surely be a main reference for anyone wishing to study partial differential operators."-- MATHEMATICAL REVIEWS "This volume focuses on linear partial differential operators with constant coefficients … . Each chapter ends with notes on the literature, and there is a large bibliography. … The binding of this softcover reprint seems quite good … . Overall, it is great to have this book back at an affordable price. It really does deserve to be described as a classic." (Fernando Q. Gouvêa, MathDL, January, 2005)Table of ContentsExistence and Approximation of Solutions of Differential Equations.- Interior Regularity of Solutions of Differential Equations.- The Cauchy and Mixed Problems.- Differential Operators of Constant Strength.- Scattering Theory.- Analytic Function Theory and Differential Equations.- Convolution Equations.

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Book SynopsisHelps you study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This title introduces some key ideas in the field, and to form a basis for further study.Trade Review"John Milnor's book provides a solid foundation and the kind of bird's eye view that perhaps only a mathematician of his caliber can offer."--William J. Satzer, MAA ReviewsTable of Contents*FrontMatter, pg. i*Table Of Contents, pg. v*List of Figures, pg. vi*Preface to the Third Edition, pg. vii*Chronological Table, pg. viii*Riemann Surfaces, pg. 1*Iterated Holomorphic Maps, pg. 39*Local Fixed Point Theory, pg. 76*Periodic Points: Global Theory, pg. 142*Structure of the Fatou Set, pg. 161*Using the Fatou Set to Study the Julia Set, pg. 174*Appendix A. Theorems from Classical Analysis, pg. 219*Appendix B. Length-Area-Modulus Inequalities, pg. 226*Appendix C. Rotations, Continued Fractions, and Rational Approximation, pg. 234*Appendix D. Two or More Complex Variables, pg. 246*Appendix E. Branched Coverings and Orbifolds, pg. 254*Appendix F. No Wandering Fatou Components, pg. 259*Appendix G. Parameter Spaces, pg. 266*Appendix H. Computer Graphics and Effective Computation, pg. 271*References, pg. 277*Index, pg. 293

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Book SynopsisThe author presents an accessible, clear introduction to dynamical systems and chaos theory, important and exciting areas that have shaped many scientific fields. While the rules governing dynamical systems are well-specified and simple, the behavior of many dynamical systems is remarkably complex.Trade Review"Feldman succeeds in introducing the reader to the world of dynamic systems and the, almost mythical, chaos that they can produce."---Adhemar Bultheel, European Mathematical Society"[A] gentle and loving introduction to dynamical systems. . . . Chaos and Dynamical Systems is a book for everyone from the layman to the expert."---David S. Mazel, MAA Reviews

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Book SynopsisThe three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and instructors. This first volume focuses on the analysis of real-valued functions of a real variable. Besides developing the basic theory it describes many applications, including a chapter on Fourier series. It also includes a Prologue in which the author introduces the axioms of set theory and uses them to construct the real number system. Volume 2 goes on to consider metric and topological spaces and functions of several variables. Volume 3 covers complex analysis and the theory of measure and integration.Trade Review'Garling is a gifted expositor and the book under review really conveys the beauty of the subject, not an easy task. [It] comes with appropriate examples when needed and has plenty of well-chosen exercises as may be expected from a textbook. As the author points out in the introduction, a newcomer may be advised, on a first reading, to skip part one and take the required properties of the ordered real field as axioms; later on, as the student matures, he/she may go back to a detailed reading of the skipped part. This is good advice.' Felipe Zaldivar, MAA Reviews'This work is the first in a three-volume set dedicated to real and complex analysis that 'mathematical undergraduates may expect to meet in the first two years or so … of analysis' … The exposition is superb: open and nontelegraphic. Highly recommended. Upper-division undergraduates and graduate students.' D. Robbins, Choice'These three volumes cover very thoroughly the whole of undergraduate analysis and much more besides.' John Baylis, The Mathematical GazetteTable of ContentsIntroduction; Part I. Prologue: The Foundations of Analysis: 1. The axioms of set theory; 2. Number systems; Part II. Functions of a Real Variable: 3. Convergent sequences; 4. Infinite series; 5. The topology of R; 6. Continuity; 7. Differentiation; 8. Integration; 9. Introduction to Fourier series; 10. Some applications; Appendix: Zorn's lemma and the well-ordering principle; Index.

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Book SynopsisComplex analysis is a beautiful subject — perhaps the single most beautiful; and striking; in mathematics. It presents completely unforeseen results that are of a dramatic; even magical; nature. This invaluable book will convey to the student its excitement and extraordinary character. The exposition is organized in an especially efficient manner; presenting basic complex analysis in around 130 pages; with about 50 exercises. The material constantly relates to and contrasts with that of its sister subject; real analysis. An unusual feature of this book is a short final chapter containing applications of complex analysis to Lie theory.Since much of the content originated in a one-semester course given at the CUNY Graduate Center; the text will be very suitable for first year graduate students in mathematics who want to learn the basics of this important subject. For advanced undergraduates; there is enough material for a year-long course or; by concentrating on the first three chapters; for one-semester course.Table of ContentsFirst Concepts; Integration Along a Contour; The Main Consequences of Cauchy's Theorem; Singularities; Conformal Mappings; Applications of Complex Analysis to Lie Theory.

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Book SynopsisIn particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant;Trade Review"This nicely written book by Thangavelu is concerned with extensions of Hardy's theorem to settings that arise from noncommutative harmonic analysis.... Each chapter contains several applications to the heat equation in various settings and ends with an extensive presentation of related topics, current research, detailed references to the literature, and lists of open problems. This makes the book an invaluable resource for graduate students and researchers in harmonic analysis and applied mathematics." —SIAM Review "…Each chapter ends with useful notes and open problems. Everything is written in sufficient detail to benefit specialized interested readers…" —MATHEMATICAL REVIEWS "The authoer discusses inthe present book the original theorem of Hardy and some of its generaliztions and its connections to noncommunitave analysis, harmonic analysis and special functions. First Hardy's theorem for the Euclidian Fourier transform is treated, and a theorem of Beurling and Hömander Subsequently Hardy's theorem is dicussed for the Fourier transfom on the Heisenberg group. finally the author discusses generaliztions of Hardy's theorem involving the Helgason Fourier transform for rank one symmetric spaces and for H-type groups. This unique book will be of great value for readers interested in this branch of analysis." ---Monatshefte für MathematikTable of Contents1 Euclidean Spaces.- 1.1 Fourier transform on L1(?n).- 1.2 Hermite functions and L2 theory.- 1.3 Spherical harmonics and symmetry properties.- 1.4 Hardy’s theorem on ?n.- 1.5 Beurling’s theorem and its consequences.- 1.6 Further results and open problems.- 2 Heisenberg Groups.- 2.1 Heisenberg group and its representations.- 2.2 Fourier transform on Hn.- 2.3 Special Hermite functions.- 2.4 Fourier transform of radial functions.- 2.5 Unitary group and spherical harmonics.- 2.6 Spherical harmonics and the Weyl transform.- 2.7 Weyl correspondence of polynomials.- 2.8 Heat kernel for the sublaplacian.- 2.9 Hardy’s theorem for the Heisenberg group.- 2.10 Further results and open problems.- 3 Symmetric Spaces of Rank 1.- 3.1 A Riemannian space associated to Hn.- 3.2 The algebra of radial functions on S.- 3.3 Spherical Fourier transform.- 3.4 Helgason Fourier transform.- 3.5 Hecke-Bochner formula for the Helgason Fourier transform.- 3.6 Jacobi transforms.- 3.7 Estimating the heat kernel.- 3.8 Hardy’s theorem for the Helgason Fourier transform.- 3.9 Further results and open problems.

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Book SynopsisThis up-to-date introduction to Griffiths' theory of period maps and period domains focusses on algebraic, group-theoretic and differential geometric aspects. The second edition has been thoroughly revised and now includes a new third section covering recent and important new developments in the field.Trade ReviewReview of previous edition: 'This book, dedicated to Philip Griffiths, provides an excellent introduction to the study of periods of algebraic integrals and their applications to complex algebraic geometry. In addition to the clarity of the presentation and the wealth of information, this book also contains numerous problems which render it ideal for use in a graduate course in Hodge theory.' Mathematical ReviewsReview of previous edition: '… generally more informal and differential-geometric in its approach, which will appeal to many readers … the book is a useful introduction to Carlos Simpson's deep analysis of the fundamental groups of compact Kähler manifolds using harmonic maps and Higgs bundles.' Burt Totaro, University of Cambridge'This monograph provides an excellent introduction to Hodge theory and its applications to complex algebraic geometry.' Gregory Pearlstein, Nieuw Archief voor WeskundeTable of ContentsPart I. Basic Theory: 1. Introductory examples; 2. Cohomology of compact Kähler manifolds; 3. Holomorphic invariants and cohomology; 4. Cohomology of manifolds varying in a family; 5. Period maps looked at infinitesimally; Part II. Algebraic Methods: 6. Spectral sequences; 7. Koszul complexes and some applications; 8. Torelli theorems; 9. Normal functions and their applications; 10. Applications to algebraic cycles: Nori's theorem; Part III. Differential Geometric Aspects: 11. Further differential geometric tools; 12. Structure of period domains; 13. Curvature estimates and applications; 14. Harmonic maps and Hodge theory; Part IV. Additional Topics: 15. Hodge structures and algebraic groups; 16. Mumford–Tate domains; 17. Hodge loci and special subvarieties; Appendix A. Projective varieties and complex manifolds; Appendix B. Homology and cohomology; Appendix C. Vector bundles and Chern classes; Appendix D. Lie groups and algebraic groups; References; Index.

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Book SynopsisThis book is based on lectures presented over many years to second and third year mathematics students in the Mathematics Departments at Bedford College, London, and King's College, London, as part of the BSc. and MSci. program. Its aim is to provide a gentle yet rigorous first course on complex analysis.Metric space aspects of the complex plane are discussed in detail, making this text an excellent introduction to metric space theory. The complex exponential and trigonometric functions are defined from first principles and great care is taken to derive their familiar properties. In particular, the appearance of π, in this context, is carefully explained.The central results of the subject, such as Cauchy's Theorem and its immediate corollaries, as well as the theory of singularities and the Residue Theorem are carefully treated while avoiding overly complicated generality. Throughout, the theory is illustrated by examples.A number of relevant results from real analysis are collected, complete with proofs, in an appendix.The approach in this book attempts to soften the impact for the student who may feel less than completely comfortable with the logical but often overly concise presentation of mathematical analysis elsewhere.Trade Review"Good bibliography; adequate index."ChoiceTable of ContentsComplex numbers; Sequences and Series; Metric Space Properties of the Complex Plane; Analytic Functions; The Complex Exponential and Trigonometric Functions; The Complex Logarithm; Complex Integration; Cauchy's Theorem; The Laurent Expansion; Singularities and Meromorphic Functions; Theory of Residues; The Argument Principle; Maximum Modulus Principle; Mobius Transformations; Harmonic Functions; Local Properties of Analytic Functions.

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Book SynopsisThe principle of local activity explains the emergence of complex patterns in a homogeneous medium. At first defined in the theory of nonlinear electronic circuits in a mathematically rigorous way, it can be generalized and proven at least for the class of nonlinear reaction-diffusion systems in physics, chemistry, biology, and brain research. Recently, it was realized by memristors for nanoelectronic device applications. In general, the emergence of complex patterns and structures is explained by symmetry breaking in homogeneous media, which is caused by local activity. This book argues that the principle of local activity is really fundamental in science, and can even be identified in quantum cosmology as symmetry breaking of local gauge symmetries generating the complexity of matter and forces in our universe. Applications are considered in economic, financial, and social systems with the emergence of equilibrium states, symmetry breaking at critical points of phase transitions and risky acting at the edge of chaos.Table of ContentsThe Local Activity Principle and the Emergence of Complexity; Local Activity and Edge of Chaos in Computer Visualization; The Local Activity Principle and the Expansion of the Universe; The Local Activity Principle and the Dynamics of Matter; The Local Activity Principle and the Evolution of Life; The Local Activity Principle and the Co-evolution of Technology; The Local Activity Principle and Innovation in the Economy and Society; The Message of the Local Activity Principle.

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