Manfred Einsiedler – författare

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence.

Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl''s polynomial equidistribution theorem, the ergodic proof of Szemeredi''s theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits.

Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.

Starting at an elementary level, the book gradually eases the reader into the complexities of higher mathematics; in particular, the formal structure of mathematical writing (definitions, theorems and proofs) is introduced in simple terms. The book covers a range of topics, from the very foundations (numbers, set theory) to basic abstract algebra (groups, rings, fields), driven throughout by the need to understand concrete equations and problems, such as determining which numbers are sums of squares. Some topics usually reserved for a more advanced audience, such as Eisenstein integers or quadratic reciprocity, are lucidly presented in an accessible way. The book also introduces the reader to open source software for computations, to enhance understanding of the material and nurture basic programming skills. For the more adventurous, a number of Outlooks included in the text offer a glimpse of possible mathematical excursions.

This book supports readers in transition from high school to university mathematics, and will also benefit university students keen to explore the beginnings of algebraic number theory. It can be read either on its own or as a supporting text for first courses in algebra or number theory, and can also be used for a topics course on Diophantine equations.

This book provides an overview of recent advances in the theory of dynamical systems, with a particular emphasis on their connections to other areas of mathematical research, including number theory, geometry, mathematical physics, complex analysis, and celestial mechanics. Compiling  the lecture notes from some of the contributions presented at the C.I.M.E. school "Modern Aspects of Dynamical Systems" held in Cetraro in August 2021, the contributions are the following: “Homogeneous dynamics and Diophantine problems” by Manfred Einsiedler, “Effective ergodic theory for translation flow” by Giovanni Forni, “Integrability and rigidity for convex billiards” by Vadim Kaloshin, “Holomorphic dynamics” by Jasmin Raissy and “Exponentially small phenomena and its role in the dynamics” by Tere Martinez-Seara.

These notes are suitable for graduate students and young researchers interested in an introduction to some of the modern research areas within the field of dynamical systems.

Dieses Buch bietet eine erste Einführung in die mathematische Theorie der dynamischen Systeme, die für Studierende des letzten Studienjahres des Bachelor Studiums und für das Master Studium geeignet ist. Aufbauend auf den Grundbegriffen der Topologischen Dynamik und der Ergodentheorie in den ersten beiden Kapiteln behandelt das dritte Kapitel den für die Ergodentheorie zentralen Begriff der Entropie, der seinen Ursprung in der statistischen Physik und der Informationstheorie hat, und der die Komplexität eines maßtheoretischen dynamischen Systems quantifiziert. Das vierte Kapitel ist ebenfalls der Entropie gewidmet, diesmal aber im Rahmen der topologischen Dynamik, bei der Entropie einen quantitativen Ausdruck für die Verformung eines kompakten metrischen Raumes durch eine stetige Transformation darstellt. Das fünfte und letzte Kapitel gibt einen kleinen Einblick in aktuelle Entwicklungen der Theorie der dynamischen Systeme mit ihren mehrparametrischen Verallgemeinerungen des klassischen Konzepts der ‚Zeitentwicklung‘ und den daraus entspringenden und zum Teil überraschenden Querverbindungen zu anderen mathematischen Disziplinen.Das in Vorlesungen erprobte Material dieses Buches kann durch eine den Interessen der Studierenden angepasste Themenauswahl wahlweise für eine ein- oder zweisemestrige Vorlesung eingesetzt werden.

This textbook provides a careful treatment of functional analysis and some of its applications in analysis, number theory, and ergodic theory.

In addition to discussing core material in functional analysis, the authors cover more recent and advanced topics, including Weyl’s law for eigenfunctions of the Laplace operator, amenability and property (T), the measurable functional calculus, spectral theory for unbounded operators, and an account of Tao’s approach to the prime number theorem using Banach algebras. The book further contains numerous examples and exercises, making it suitable for both lecture courses and self-study.

Functional Analysis, Spectral Theory, and Applications is aimed at postgraduate and advanced undergraduate students with some background in analysis and algebra, but will also appeal to everyone with an interest in seeing how functional analysis can be applied to other parts of mathematics.