Jürgen Jost is a German mathematician whose work spans a wide range of topics in mathematics and theoretical physics. His research often explores the connections between geometry, analysis, and probability, focusing on the study of complex systems and their behavior. Jost is recognized for his contributions to areas such as Riemannian geometry, differential geometry, and statistical physics. His approach to mathematics is characterized by a pursuit of deeper understanding of the fundamental principles governing natural phenomena.
Focusing on the conceptual and abstract foundations of mathematics, this book emphasizes the development of structural thinking. It employs specific mathematical structures to highlight relationships and common features, offering a clear and insightful approach. The text is designed to motivate and explain complex ideas through examples, allowing for the omission of many technical proofs. This makes it accessible for readers seeking to deepen their understanding of mathematical concepts without getting bogged down in intricate details.
Focusing on the intersection of mathematics and biology, this book equips readers with essential mathematical tools needed in modern biological research. It covers a wide range of mathematical concepts, including stochastic processes and pattern formation, and illustrates these ideas through biological examples that span molecular, evolutionary, and ecological contexts.
„Geometry and Physics“ addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jürgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective.
Our aim is to introduce, explain, and discuss the fundamental problems, ideas, concepts, results, and methods of the theory of dynamical systems and to show how they can be used in speci? c examples. We do not intend to give a comprehensive overview of the present state of research in the theory of dynamical systems, nor a detailed historical account of its development. We try to explain the important results, often neglecting technical re? nements 1 and, usually, we do not provide proofs. One of the basic questions in studying dynamical systems, i. e. systems that evolve in time, is the construction of invariants that allow us to classify qualitative types of dynamical evolution, to distinguish between qualitatively di? erent dynamics, and to studytransitions between di? erent types. Itis also important to ? nd out when a certain dynamic behavior is stable under small perturbations, as well as to understand the various scenarios of instability. Finally, an essential aspect of a dynamic evolution is the transformation of some given initial state into some ? nal or asymptotic state as time proceeds. Thetemporalevolutionofadynamicalsystemmaybecontinuousordiscrete, butitturnsoutthatmanyoftheconceptstobeintroducedareusefulineither case.
Modern and systematic treatment of main approaches; Several additions have been made to the German edition, most notably coverage of eigenvalues and expansions; Emphasis on methods relevant for both linear and nonlinear equations; Contains chapter summaries, detailed illustrations and numerous exercises
This text on Riemannian geometry and geometric analysis is written for graduates and reseachers and includes material new to the late 1990s on Ginzburg-Landau, Seibert-Witten functionals, spin geometry and Dirac operators.
An introduction to advanced analysis, this textbook blends modern presentation with concrete examples and applications - especially in the areas of calculus of variations and partial differential equations. Banach space, Lebesgue integration theory and Sobolev space theory are all discussed.
Focusing on nonpositive curvature, the book explores both geometric and analytic dimensions, beginning with Riemannian examples and rigidity theorems. It delves into generalized concepts of nonpositive curvature within metric geometry, referencing Alexandrov and Busemann's frameworks. Additionally, the text addresses the theory of harmonic maps that take values in spaces characterized by these curvature properties, providing a comprehensive understanding of the topic.
This text explores the connection of Reimann surfaces with other fields of mathematics. It is intended as an introduction to contemporary mathematics and it includes background material from algebraic topology, differential geometry, the calculus of variations, elliptic PDE, and algebraic geometry.
