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Partial Differential Equations
This textbook is intended for students who wish to obtain an introduction to the theory of partial di?erential equations (PDEs, for short), in particular, those of elliptic type. Thus, it does not o?er a comprehensive overview of the whole ?eld of PDEs, but tries to lead the reader to the most important methods and central results in the case of elliptic PDEs. The guiding qu- tion is how one can ?nd a solution of such a PDE. Such a solution will, of course, depend on given constraints and, in turn, if the constraints are of the appropriate type, be uniquely determined by them. We shall pursue a number of strategies for ?nding a solution of a PDE; they can be informally characterized as follo...
On a Free Boundary Problem for Embedded Minimal Surfaces and Instability Theorems for Manifolds with Positive Isotropic Curvature
In this thesis, we describe a min-max construction of embedded minimal surfaces satisfying the free boundary condition in any compact 3-manifolds with boundary. We also prove the instability of minimal surfaces of certain conformal type in 4- manifolds with positive isotropic curvature. Given a compact 3-manifold M with boundary [d̳]M, consider the problem of find- ing an embedded minimal surface [Sigma] which meets [d̳]M orthogonally along [d̳][Sigma]. These surfaces are critical points to the area functional with respect to variations preserving [delta]M. We will use a min-max construction to construct such a free boundary solution and prove the regularity of such solution up to the fre...
Riemannian Geometry and Geometric Analysis
This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. The previous edition already introduced and explained the ideas of the parabolic methods that had found a spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discussed further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry. The 6th edition includes a systematic treatment of eigenvalues of Riemannian manifolds and several other additions. Also, the entire material has been reorganized in order to improve the coherence of the b...
Geometry and Physics
"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.
Stories Behind Theorems
Mathematicians are often seen through a lens of stereotypes: logical, detached thinkers who live in a world of abstract concepts, numbers, and formulas. These perceptions can make mathematics—and mathematicians—seem distant, unapproachable, or even intimidating. But they only scratch the surface. Mathematicians are as diverse, vibrant, and human as anyone else, with personal stories, creative pursuits, struggles, and achievements that shape both their lives and their work. This book brings together a selection of interviews that aim to go beyond the stereotypes and show the human side of mathematics. Each conversation offers a glimpse into the unique experiences, challenges, and passions of a different mathematician. These stories touch on the pursuit of knowledge, the balance between personal and professional life, reflections on mental health, and the central role of creativity in their work. Together, these voices show that mathematics is not only about logic and precision, but it is also shaped by diverse and often surprising perspectives.
Dynamical Systems
Breadth of scope is unique Author is a widely-known and successful textbook author Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas No technical proofs, but an introduction to the whole field that is based on the specific analysis of carefully selected examples Includes a section on cellular automata
Tsing Hua Lectures on Geometry & Analysis
A collection of lectures given by the author and numerous contributors at Tsing Hua University, Taiwan, in 1990-1. Topics covered include: energy in general relativity; existence and convergence of solutions; closed geodesics; and mean curvature evolution.
