Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers.

Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

One of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This new edition of Wilson Sutherland's classic text introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. The language of metric and topological spaces is established with continuity as the motivating concept. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. The discussion develops to cover connectedness, compactness and completeness, a trio widely used in the rest of mathematics.
Topology also has a more geometric aspect which is familiar in popular expositions of the subject as rubber-sheet geometry', with pictures of Mobius bands, doughnuts, Klein bottles and the like; this geometric aspect is illustrated by describing some standard surfaces, and it is shown how all this fits into the same story as the more analytic developments.
The book is primarily aimed at second- or third-year mathematics students. There are numerous exercises, many of the more challenging ones accompanied by hints, as well as a companion website, with further explanations and examples as well as material supplementary to that in the book."

This book explains the notion of Brakke's mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke's mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke's existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard's regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

Mumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book is the first comprehensive exploration of Mumford-Tate groups and domains. Containing basic theory and a wealth of new views and results, it will become an essential resource for graduate students and researchers.

Although Mumford-Tate groups can be defined for general structures, their theory and use to date has mainly been in the classical case of abelian varieties. While the book does examine this area, it focuses on the nonclassical case. The general theory turns out to be very rich, such as in the unexpected connections of finite dimensional and infinite dimensional representation theory of real, semisimple Lie groups. The authors give the complete classification of Hodge representations, a topic that should become a standard in the finite-dimensional representation theory of noncompact, real, semisimple Lie groups. They also indicate that in the future, a connection seems ready to be made between Lie groups that admit discrete series representations and the study of automorphic cohomology on quotients of Mumford-Tate domains by arithmetic groups. Bringing together complex geometry, representation theory, and arithmetic, this book opens up a fresh perspective on an important subject.

A quantity can be made smaller and smaller without it ever vanishing. This fact has profound consequences for science, technology, and even the way we think about numbers. In this book, we will explore this idea by moving at an easy pace through an account of elementary real analysis and, in particular, will focus on numbers, sequences, and series. Almost all textbooks on introductory analysis assume some background in calculus. This book doesn't and, instead, the emphasis is on the application of analysis to number theory. The book is split into two parts. Part 1 follows a standard university course on analysis and each chapter closes with a set of exercises. Here, numbers, inequalities, convergence of sequences, and infinite series are all covered. Part 2 contains a selection of more unusual topics that aren't usually found in books of this type. It includes proofs of the irrationality of e and , continued fractions, an introduction to the Riemann zeta function, Cantor's theory of the infinite, and Dedekind cuts. There is also a survey of what analysis can do for the calculus and a brief history of the subject. A lot of material found in a standard university course on "real analysis" is covered and most of the mathematics is written in standard theorem-proof style. However, more details are given than is usually the case to help readers who find this style daunting. Both set theory and proof by induction are avoided in the interests of making the book accessible to a wider readership, but both of these topics are the subjects of appendices for those who are interested in them. And unlike most university texts at this level, topics that have featured in popular science books, such as the Riemann hypothesis, are introduced here. As a result, this book occupies a unique position between a popular mathematics book and a first year college or university text, and offers a relaxed introduction to a fascinating and important branch of mathematics.

Das Lehrbuch vermittelt solides Basiswissen zu den thematischen Schwerpunkten Produktmasse, Fourier-Transformation, Transformationsformel, Konvergenzbegriffe, absolute Stetigkeit und Masse auf topologischen Raumen. Hoehepunkte sind die Herleitung des Riesz'schen Darstellungssatzes und der Beweis der Existenz und Eindeutigkeit des Haar'schen Masses. Der Band enthalt ferner mathematikhistorische Ausfluge und Kurzportrats von Mathematikern, die zum Thema des Buchs wichtige Beitrage geliefert haben, sowie zahlreiche UEbungsaufgaben zur Vertiefung des Stoffs.

This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.

Reellwertige Funktionen von mehreren reellen Veranderlichen.- Differentialrechnung vektorwertiger Funktionen.- Mehrdimensionale Integration.- Flachen und Flachenintegrale.- Stammfunktionen und Wegunabhangigkeit von Kurven- und Flachenintegralen.- Integralsatze von Gauss und Stokes.- Gewoehnliche Differentialgleichungen.

Dieses Buch bietet Loesungen zu den 240 Aufgaben aus dem Buch "Mathematik fur Ingenieure und Naturwissenschaftler - Band 2: Analysis in R^n und gewoehnliche Differentialgleichungen". Die Loesungen sind detailliert und verstandlich ausgearbeitet, bei einigen Aufgaben werden alternative Loesungswege vorgestellt und verglichen. Bedingt durch das breite Aufgabenspektrum eignet sich dieses Aufgaben- und Loesungsbuch fur diverse Studiengange. Neben den Studierenden der Ingenieurwissenschaften und technisch-physikalisch orientierten Studiengange profitieren auch in besonderer Weise Lehramtsstudierende und Studierende des Faches Mathematik von der Aufgabenvielfalt.

Fundamentals of Mathematical Analysis explores real and functional analysis with a substantial component on topology. The three leading chapters furnish background information on the real and complex number fields, a concise introduction to set theory, and a rigorous treatment of vector spaces. Fundamentals of Mathematical Analysis is an extensive study of metric spaces, including the core topics of completeness, compactness and function spaces, with a good number of applications. The later chapters consist of an introduction to general topology, a classical treatment of Banach and Hilbert spaces, the elements of operator theory, and a deep account of measure and integration theories. Several courses can be based on the book. This book is suitable for a two-semester course on analysis, and material can be chosen to design one-semester courses on topology or real analysis. It is designed as an accessible classical introduction to the subject and aims to achieve excellent breadth and depth and contains an abundance of examples and exercises. The topics are carefully sequenced, the proofs are detailed, and the writing style is clear and concise. The only prerequisites assumed are a thorough understanding of undergraduate real analysis and linear algebra, and a degree of mathematical maturity.

This 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.

Lively prose and imaginative exercises draw the reader into this unique introductory real analysis textbook. Motivating the fundamental ideas and theorems that underpin real analysis with historical remarks and well-chosen quotes, the author shares his enthusiasm for the subject throughout. A student reading this book is invited not only to acquire proficiency in the fundamentals of analysis, but to develop an appreciation for abstraction and the language of its expression. In studying this book, students will encounter: the interconnections between set theory and mathematical statements and proofs; the fundamental axioms of the natural, integer, and real numbers; rigorous -N and - definitions; convergence and properties of an infinite series, product, or continued fraction; series, product, and continued fraction formulae for the various elementary functions and constants. Instructors will appreciate this engaging perspective, showcasing the beauty of these fundamental results.

The 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 II goes on to consider metric and topological spaces and functions of several variables. Volume III covers complex analysis and the theory of measure and integration.

Die Entwicklung, in welcher sich die Theorie der reellen Funktionen seit einiger Zeit befindet, betrifft vor allem die allgemeinen Begriffe. Besonders die Idee der Ordnung mit allen ihren Spielarten, wie sie etwa in den Strukturen des Filters, des Verbandes, des Somenringes und der Ortsfunktionen gepragt worden ist, fuhrte in steigendem Masse zu einer Umgestaltung aller Teile der Theorie. Diese Entwicklung kann noch nicht als abgeschlossen angesehen werden; trotzdem wurde versucht, sie in diesem Buch zu berucksichtigen, zu dem Ausmass allerdings, wie es mir ursprunglich vorschwebte, ist es nicht gekommen. Verspatet erst wurde mir die einschlagige Literatur zuganglich, und ausserdem ergab es sich, dass der klassische Tatsachenbestand, der trotz aller neuen Be- griffsbildungen immer noch den eigentlichen Schatz der Theorie aus- macht, letzthin nicht vernachlassigt werden durfte. Dass auf den fol- genden 400 Seiten keine erschoepfende Behandlung des Gesamtgebietes moeglich war, ist bei der Weite desselben nicht verwunderlich. So fehlt insbesondere eine eingehende Behandlung der Theorien der Differen- tiation der additiven Mengenfunktionen, der Oberflachenintegrale, des DENJoyschen Integrals, der CARATHEODoRYschen Ortsfunktionen und der SCHWARTzschen Distributionen; das Literaturverzeichnis am Ende des Buches mag ein kleiner Luckenbusser dafur sein. Wegen der hier behandelten Gegenstande selbst aber verweise ich auf den nachfolgenden UEberblick.

Classical Sobolev spaces, based on Lebesgue spaces on an underlying domain with smooth boundary, are not only of considerable intrinsic interest but have for many years proved to be indispensible in the study of partial differential equations and variational problems. Many developments of the basic theory since its inception arise in response to concrete problems, for example, with the (ubiquitous) sets with fractal boundaries.

The theory will probably enjoy substantial further growth, but even now a connected account of the mature parts of it makes a useful addition to the literature. Accordingly, the main themes of this book are Banach spaces and spaces of Sobolev type based on them; integral operators of Hardy type on intervals and on trees; and the distribution of the approximation numbers (singular numbers in the Hilbert space case) of embeddings of Sobolev spaces based on generalised ridged domains.

This timely book will be of interest to all those concerned with the partial differential equations and their ramifications. A prerequisite for reading it is a good graduate course in real analysis.

Die nachfolgenden Tabellen stellen eine Sammlung von Integralen der folgenden Form dar. 00 (1 ) g(y) = f I(x) cos(xy)dx (Erstes Kapitel) 0 00 (2) g (y) = f I (x) sin (x y) d x (Zweites Kapitel) 0 00 ixy g(y) = Jt(x) e dx (Drittes Kapitel). (3 ) -00 Die Funktion g(y) in (1), (2) und (3) wird der Reihe nach als FOURIER- Kosinus-, FOURIER-Sinus-, und exponentielle FOURIER-Transformation der Funktion I (x) bezeichnet. Unter gewissen Bedingungen [s. z. B. eines der im Literaturverzeichnis unter a) aufgefiihrten WerkeJ gelten die (1), (2) und (3) entsprechenden Umkehrformeln 00 (1 a) I(x) = . f g(y) cos(xy) dy o 00 (2a) I(x) = J g(y) sin(xy) dy o 00 I(x) = . LJg(y) e-ixYdy. 2n -00 Offensichtlich geht das Formelpaar (3), (3a) in (1), (1 a) oder (2), (2a) iiber, je nachdem I(x) gerade oder ungerade ist. In den Tabellen sind Parameter die durch lateinische Buchstaben bezeichnet sind, wenn nicht anders vermerkt, als positiv und reell vorausgesetzt, wobei fUr die Beispiele im dritten Kapitel der Parameter yauch negative Werte annimmt. In den meisten Fallen ist der Giiltigkeitsbereich eines Formel- paares fUr komplexe Werte dieser GraBen sofort ersichtlich. Griechische Buchstaben bedeuten komplexe Parameter innerhalb des angegebenen Giiltigkeitsbereiches. In einigen Fallen ist die Funktion g (y) nur iiber einen Teilbereich von y angegeben. Dies bedeutet, daB sich g (y) fUr den restlichen Bereich nicht in einfacher Form angeben liiBt.

There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.

This book provides a rigorous treatment of multivariable differential and integral calculus. Implicit function theorem and the inverse function theorem based on total derivatives is explained along with the results and the connection to solving systems of equations. There is an extensive treatment of extrema, including constrained extrema and Lagrange multipliers, covering both first order necessary conditions and second order sufficient conditions. The material on Riemann integration in n dimensions, being delicate by its very nature, is discussed in detail. Differential forms and the general Stokes' Theorem are expounded in the last chapter. With a focus on clarity rather than brevity, this text gives clear motivation, definitions and examples with transparent proofs. Much of the material included is published for the first time in textbook form, for example Schwarz' Theorem in Chapter 2 and double sequences and sufficient conditions for constrained extrema in Chapter 4. A wide selection of problems, ranging from simple to more challenging, are included with carefully formed solutions. Ideal as a classroom text or a self study resource for students, this book will appeal to higher level undergraduates in Mathematics.

Diese Einfuhrung in die Analysis orientiert sich an der historischen Entwicklung: Die ersten zwei Kapitel schlagen den Bogen von historischen Berechnungsmethoden zu unendlichen Reihen, zur Differential- und Integralrechnung und zu Differentialgleichungen. Die Etablierung einer mathematisch stringenten Denkhaltung im 19. Jahrhundert fur ein und mehrere Variablen ist Thema der darauffolgenden Kapitel. Viele Beispiele, Berechnungen und Bilder machen den Band zu einem Lesevergnugen fur Studierende, fur Lehrer und fur Wissenschaftler.

Wer Mathematik liebt, findet sie spannend und aufregend. F r viele Menschen ist Mathematik allerdings ein Buch mit sieben Siegeln. Dieses Buch schl gt Br cken in das Reich der Mathematik. Es richtet sich an Leser jeden Alters und jeder Vorbildung. Gymnasiallehrer finden eine reiche Auswahl an Beispielen, Studenten bietet es Orientierung, und Dozenten werden sich an den Feinheiten der Darstellung zweier Meister ihres Faches erfreuen.

The aim of the present book is a uni?ed representation of some recent results in geometric function theory together with a consideration of their historical sources. These results are concerned with functions f, holomorphic or meromorphic in a domain ? in the extended complex planeC. The only additional condition we impose on these functions is the condition that the range f(?) is contained in a given domain ??C.Thisfactwillbedenotedby f? A(?,?). We shall describe (n) how one may get estimates for the derivatives|f (z )|,n?N,f ? A(?,?), 0 dependent on the position of z in ? and f(z)in?. 0 0 1.1 Historical remarks The beginning of this program may be found in the famous article [125] of G. Pick. There, he discusses estimates for the MacLaurin coe?cients of functions with positive real part in the unit disc found by C. Carath' eodory in [52]. Pick tells his readers that he wants to generalize Carath' eodory's estimates such that the special role of the expansion point at the origin is no longer important. For the convenience of our readers we quote this sentence in the original language: Durch lineare Transformation von z oder, wie man sagen darf, durch kreis- ometrische Verallgemeinerung, kann man die Sonderstellung des Wertes z=0 wegscha?en, so dass sich Relationen fur .. die Di?erentialquotienten von w an - liebiger Stelle ergeben. The ?rst great success of this program was G. Pick's theorem, as it is called by Carath' eodory himself, compare [54], vol II, 286-289.

Even the simplest singularities of planar curves, e.g. where the curve crosses itself, or where it forms a cusp, are best understood in terms of complex numbers. The full treatment uses techniques from algebra, algebraic geometry, complex analysis and topology and makes an attractive chapter of mathematics, which can be used as an introduction to any of these topics, or to singularity theory in higher dimensions. This book is designed as an introduction for graduate students and draws on the author's experience of teaching MSc courses; moreover, by synthesising different perspectives, he gives a novel view of the subject, and a number of new results.

Les deux premiers volumes sont consacrA(c)s aux fonctions dans R ou C, y compris la thA(c)orie A(c)lA(c)mentaire des sA(c)ries et intA(c)grales de Fourier et une partie de celle des fonctions holomorphes. L'exposA(c) non strictement linA(c)aire, combine indications historiques et raisonnements rigoureux. Il montre la diversitA(c) des voies d'accA]s aux principaux rA(c)sultats afin de familiariser le lecteur avec les mA(c)thodes de raisonnement et idA(c)es fondamentales plutAt qu'avec les techniques de calcul, point de vue utile aussi aux personnes travaillant seules.
Les volumes 3 et 4 traitent principalement des fonctions analytiques (thA(c)orie de Cauchy, thA(c)orie analytique des nombres et fonctions modulaires), ainsi que du calcul diffA(c)rentiel sur les variA(c)tA(c)s, avec un exposA(c) de l'intA(c)grale de Lebesgue, en suivant d'assez prA]s le cA(c)lA]bre cours donnA(c) longtemps par l'auteur A l'UniversitA(c) Paris 7.
On reconnaA(R)tra dans ce nouvel ouvrage le style inimitable de l'auteur, et pas seulement par son refus de l'A(c)criture condensA(c)e en usage dans ce nombreux manuels.

Ce 4A]me volume de l'ouvrage Analyse mathA(c)matique initiera le lecteur A l'analyse fonctionnelle (intA(c)gration, espaces de Hilbert, analyse harmonique en thA(c)orie des groupes) et aux mA(c)thodes de la thA(c)orie des fonctions modulaires (sA(c)ries L et theta, fonctions elliptiques, usage de l'algA]bre de Lie de SL2). Tout comme pour les volumes 1 A 3, on reconnaA(R)tra ici encore, le style inimitable de l'auteur et pas seulement par son refus de l'ecriture condensA(c)e en usage dans de nombreux manuels. Mariant judicieusement les mathA(c)matiques dites 'modernes' et' classiques', la premiA]re partie (IntA(c)gration) est d'utilitA(c) universelle tandis que la seconde oriente le lecteur vers un domaine de recherche spA(c)cialisA(c) et trA]s actif, avec de vastes gA(c)nA(c)ralisations possibles.

Ce vol. III expose la thA(c)orie classique de Cauchy dans un esprit orientA(c) bien davantage vers ses innombrables utilisations que vers une thA(c)orie plus ou moins complA]te des fonctions analytiques. On montre ensuite comment les intA(c)grales curvilignes A la Cauchy se gA(c)nA(c)ralisent A un nombre quelconque de variables rA(c)elles (formes diffA(c)rentielles, formules de type Stokes). Les bases de la thA(c)orie des variA(c)tA(c)s sont ensuite exposA(c)es, principalement pour fournir au lecteur le langage "canonique" et quelques thA(c)orA]mes importants (changement de variables dans les intA(c)grales, A(c)quations diffA(c)rentielles). Un dernier chapitre montre comment on peut utiliser ces thA(c)ories pour construire la surface de Riemann compacte d'une fonction algA(c)brique, sujet rarement traitA(c) dans la littA(c)rature non spA(c)cialisA(c)e bien que n'A(c)xigeant que des techniques A(c)lA(c)mentaires. Un volume IV exposera, outre, l'intA(c)grale de Lebesgue, un bloc de mathA(c)matiques spA(c)cialisA(c)es vers lequel convergera tout le contenu des volumes prA(c)cA(c)dents: sA(c)ries et produits infinis de Jacobi, Riemann, Dedekind, fonctions elliptiques, thA(c)orie classique des fonctions modulaires et la version moderne utilisant la structure de groupe de Lie de SL(2, R).