A theory is the more impressive, the simpler are its premises, the more distinct are the things it connects, and the broader is its range of applicability. Albert Einstein There are two different ways of teaching mathematics, namely, (i) the systematic way, and (ii) the application-oriented way. More precisely, by (i), I mean a systematic presentation of the material governed by the desire for mathematical perfection and completeness of the results. In contrast to (i), approach (ii) starts out from the question "What are the most important applications?" and then tries to answer this question as quickly as possible. Here, one walks directly on the main road and does not wander into all the nice and interesting side roads. The present book is based on the second approach. It is addressed to undergraduate and beginning graduate students of mathematics, physics, and engineering who want to learn how functional analysis elegantly solves mathematical problems that are related to our real world and that have played an important role in the history of mathematics. The reader should sense that the theory is being developed, not simply for its own sake, but for the effective solution of concrete problems. viii Preface Our introduction to applied functional analysis is divided into two parts: Part I: Applications to Mathematical Physics (AMS Vol. 108); Part II: Main Principles and Their Applications (AMS Vol. 109). A detailed discussion of the contents can be found in the preface to AMS Vol. 108.
This Edited Volume is based on a workshop on “Mathematical and Physical - pects of Quantum Gravity” held at the Heinrich-Fabri Institute in Blaubeuren st (Germany) from July 28th to August 1 , 2005. This workshop was the succ- sor of a similar workshop held at the same place in September 2003 on the issue of “Mathematical and Physical Aspects of Quantum Field Theories”. Both wo- shops were intended to bring together mathematicians and physicists to discuss profoundquestionswithin the non-emptyintersectionofmathematics andphysics. The basic idea of this series of workshops is to cover a broad range of di?erent approaches (both mathematical and physical) to a speci?c subject in mathema- cal physics. The series of workshops is intended, in particular, to discuss the basic conceptual ideas behind di?erent mathematical and physical approaches to the subject matter concerned. The workshop on which this volume is based was devoted to what is c- monly regarded as the biggest challenge in mathematical physics: the “quanti- tion of gravity”. The gravitational interaction is known to be very di?erent from the known interactions like, for instance, the electroweak or strong interaction of elementary particles. First of all, to our knowledge, any kind of energy has a gravitational coupling. Second, since Einstein it is widely accepted that gravity is intimately related to the structure of space-time. Both facts have far reaching consequences for any attempt to develop a quantum theory of gravity.
For more than 70 years, quantum field theory (QFT) can be seen as a driving force in the development of theoretical physics. The developed ideas and techniques of QFT have been successfully applied, in particular, within the phenomenological description of particle physics and solid state physics. Equally fascinating is the fruitful impact which QFT had in rather remote areas of mathematics, like Gromov-Witten and Donaldson-Witten invariants of low dimensional manifolds and for modular forms in relation to string theory. More recent developments in QFT also attack the problem to formulate a quantum version of gravity. However, there is no 'QFT as such', but instead there are only various mathematical approaches, aiming to make the basic ideas of QFT more rigorous. Such a rigorous understanding seems indispensable, in particular, to get a better understanding of how a physically reasonable quantum theory of gravity may look like.

The present book features some of the different approaches, different physical viewpoints and techniques used to make the notion of quantum field theory more precise. This concerns algebraic, analytic, geometric, and stochastic aspects. For example, there will be discussed deformation theory, and the holographic AdS/CFT correspondence. The book also contains more recent developments like the use of category theory and topos theoretic methods to describe QFT. This volume emerged from the 3rd 'Blaubeuren Workshop: Recent Developments in Quantum Field Theory', held in July 2007 at the Max Planck Institute of Mathematics in the Sciences in Leipzig/Germany. All of the contributions to the volume are peer reviewed and committed to the idea of this workshop series: 'To bring together outstanding experts working in the field of mathematics and physics to discuss in an open atmosphere the fundamental questions at the frontier of theoretical physics'.

Mit dem "TEUBNER-TASCHENBUCH der Mathematik· Teil II" liegt eine völlig neube arbeitete und wesentlich erweiterte Auflage der bisherigen "Ergänzenden Kapitel zum Taschenbuch der Mathematik von I. N. Bronstein und K. A. Semendjajew" vor, die 1990 in sechster Auflage im Verlag B. G. Teubner in Leipzig erschienen sind. Der "Bronstein", das unentbehrliche Nachschlagewerk für Generationen von Studenten, Lehrern und Praktikern, wurde 1957 von V. Ziegler aus dem Russischen übersetzt und erstmals 1958 in deutscher Sprache - erweitert um die Abschnitte "Variationsrechnung" und "Integralgleichungen" -im Verlag B. G. Teubner in Leipzig veröffentlicht. Unter der Herausgabe von G. Grosche und V. Ziegler erschien 1979 in enger Abstim mung mit den Autoren der ursprünglichen Fassung die 19. , völlig überarbeitete Auflage als Gemeinschaftsausgabe der Verlage Nauka und Teubner. In diese Leipziger Neubear beitung wurden neue Teilgebiete aufgenommen, wie Grundbegriffe der mathematischen Logik, Maßtheorie und Lebesgue-Stieltjes-Integral, Tensorrechnung, lineare Optimierung, nichtlineare Optimierung, dynamische Optimierung, Graphentheorie, Spieltheorie, Nu merik und Funktionalanalysis. Manche Abschnitte mußten erheblich erweitert werden, z. B. Wahrscheinlichkeitsrechnung, mathematische Statistik, Fourier-Analyse und Laplace Transformation. Da bei dieser Zielstellung der Umfang des Werkes nicht annähernd in den Grenzen des ursprünglichen Taschenbuches gehalten werden konnte, die Handlichkeit aber möglichst erhalten bleiben sollte, kamen Verfasser der ursprünglichen und Herausgeber der neuen Fassung zu dem Entschluß, die weiterführenden Kapitel 8 bis 11 herauszulösen und in einem Ergänzungsband zusammenzufassen.
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