Terence Tao Libri

This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

This is part two of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.

Authored by a leading name in mathematics, this engaging and clearly presented text leads the reader through the tactics involved in solving mathematical problems at the Mathematical Olympiad level. With numerous exercises and assuming only basic mathematics, this text is ideal for students of 14 years and above in pure mathematics.

Können wir mit Mathematik die Vorhersage und den Ausgang der amerikanischen Präsidentschaftswahl 2008 verstehen? Gab es nicht jede Menge unentschlossener Wähler? Wie wirken sich spezielle Eigenschaften von Systemkomponenten auf das Ganze aus? Welches sind die universellen Gesetze in Natur und Gesellschaft? Was haben 150 Jahre alte rein mathematische Probleme mit Atomen zu tun? Warum geht am Aktienmarkt manchmal alles gewaltig schief? Der Fields-Medaillen-Gewinner Terence Tao geht genau diesen Fragen über Universalität und Komplexität nach. Er zeigt, dass einfache statistische Gesetze für viele Vorgänge gelten und wo deren Grenzen liegen. Tao hat mit 'E pluribus unum: From Complexity, Universality' seinen ersten populären Artikel verfasst. Die Fotos von Jochem Berlemann lassen uns vergessen, dass Mathematik eine abstrakte, mühsame und wenig anschauliche Wissenschaft ist. Sie schaffen eine Atmosphäre der Neugier und wetteifern mit weiteren Illustrationen und Bildern um unsere Aufmerksamkeit. Das Buch wendet sich an interessierte Leserinnen und Leser aus Mathematik, Physik und Naturwissenschaft.