Diese Lehrbuchreihe richtet sich an fortgeschrittene Studenten und Doktoranden am Anfang ihrer Karriere. Sie deckt das gesamte Spektrum der reinen Mathematik ab, ebenso wie Themen der angewandten Mathematik und mathematischen Physik, die einen erheblichen Einsatz moderner mathematischer Methoden erfordern. Auch weniger standardmäßige, aber aktuell relevante Themen finden Aufnahme in dieser Reihe.
Focusing on recent advancements in permutation groups, this book serves as an introductory guide tailored for beginning graduate students. It presents key concepts and developments in the field, making complex topics accessible for those new to the subject. Through clear explanations and structured content, readers will gain a solid foundation in permutation group theory.
Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.
Focusing on the classical aspects of Banach space theory, this short course covers three key topics: Schauder bases, Lp spaces, and C(K) spaces. It highlights the postwar revival of the field led by notable mathematicians such as James and Lindenstrauss, showcasing their impactful results. A foundational knowledge of functional analysis and some exposure to abstract measure theory is required, along with a basic understanding of topology, although an appendix is provided for necessary topological concepts.
Focusing on the foundations of logic and set theory, this book offers a distinctive approach to axiomatization. It presents fundamental concepts in a clear manner, making complex ideas accessible to readers. By exploring the interplay between logic and set theory, it provides valuable insights for those interested in the philosophical and mathematical underpinnings of these disciplines.
This book provides a quick yet detailed introduction to set theory and forcing, building the reader's intuition about it as well as rigorousness. Part II discusses contemporary issues in the theory of forcing, including previously unpublished results and open questions.
This book offers a comprehensive exploration of representation theorems, presenting direct proofs for two distinct classes of Hardy spaces. It delves into the theoretical underpinnings and practical implications of these theorems, making complex concepts accessible for readers. The clear explanations and structured approach aim to enhance understanding of Hardy spaces and their applications in mathematical analysis.
Focusing on geometric group theory, this book offers an engaging and approachable introduction tailored for advanced undergraduates. It combines clarity with depth, making complex concepts accessible while encouraging a deeper understanding of the subject.
The book focuses on modern set theory techniques that are particularly relevant to various fields of pure mathematics. It explores how these methods can be applied to enhance understanding and problem-solving across different mathematical disciplines, making it a valuable resource for mathematicians seeking to deepen their knowledge and skills in set theory and its applications.
Classical number theory is developed from scratch leading to geometric discrepancy theory, with Fourier analysis introduced along the way.
The author develops the combinatorics of Young tableaux and shows them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The intuitive presentation and numerous exercises make the book suitable for beginning graduate students as well as researchers.
Rooted in a course from the Massachusetts Institute of Technology, this book delves into advanced concepts and methodologies relevant to its subject matter. It offers a unique blend of theoretical insights and practical applications, making it an essential resource for students and professionals alike. The content is designed to enhance understanding and foster critical thinking, providing readers with the tools needed to navigate complex topics effectively.
This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time.
This text is ideal for advanced undergraduate or beginning graduate students. The author first develops the necessary background in probability theory and Markov chains before using it to study a range of randomized algorithms with important applications in optimization and other problems in computing. The book will appeal not only to mathematicians, but to students of computer science who will discover much useful material. This clear and concise introduction to the subject has numerous exercises that will help students to deepen their understanding.
This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.
Focusing on ergodic theory, this book offers a comprehensive introduction to equilibrium states, structured around a one-semester course format. It is designed to be self-contained, making complex concepts accessible to readers. The material is presented in a clear manner, ideal for those seeking to understand the foundational principles of ergodic theory without prior extensive knowledge.
The book provides a comprehensive overview of computational techniques for solving diophantine equations, focusing on the mathematical concepts and algorithms involved. It explores various methods, offering insights into their applications and effectiveness in tackling these equations. The text is designed for both researchers and students, aiming to enhance understanding of this complex area in number theory.
Focusing on permutation groups, this text highlights recent advancements influenced by the Classification of Finite Simple Groups and its connections to logic and combinatorics. It introduces relevant computer algebra systems capable of handling large groups and includes sketch proofs of key theorems alongside practical examples. Designed for beginning graduate students and experts from various fields, the content is based on a short course held at the EIDMA institute in Eindhoven, making complex topics accessible to a broader audience.
The theory of semigroups of operators is a topic with great intellectual beauty and wide-ranging applications. This graduate-level introduction presents the essential elements of the theory, introducing the key notions and establishing the central theorems. A mixture of applications are included and further development directions are indicated.
The book is a self-contained introduction to the results and computational methods in classical invariant theory. It relies on minimal mathematical prerequisites, and can be read by advanced undergraduate or graduate students. A variety of innovations will also make the book of interest to specialists and researchers.
Recent years have seen rapid progress in the field of approximate groups, with the emergence of a varied range of applications. Written by a leader in the field, this text for both beginning graduate students and researchers collects, for the first time in book form, the main concepts and techniques into a single, self-contained introduction.