低维系统的几何与拓扑

English | 18 October 2024 | ISBN-10: 3031595009 | 181 pages| Epub PDF (True) | 13 MB

This book introduces the field of topology, a branch of mathematics that explores the properties of geometric space, with a focus on low-dimensional systems. The authors discuss applications in various areas of physics. The first chapters of the book cover the formal aspects of topology, including classes, homotopic groups, metric spaces, and Riemannian and pseudo-Riemannian geometry. These topics are essential for understanding the theoretical concepts and notations used in the next chapters of the book. The applications encompass defects in crystalline structures, space topology, spin statistics, Braid group, Chern-Simons field theory, and 3D gravity, among others. This self-contained book provides all the necessary additional material for both physics and mathematics students. The presentation is enriched with examples and exercises, making it accessible for readers to grasp the concepts with ease. The authors adopt a pedagogical approach, posing many unsolved questions in simple situations that can serve as challenging projects for students. Suitable for a one-semester postgraduate level course, this text is ideal for teaching purposes.

中文|2024年10月18日|ISBN-10:3031595009|181页|Epub PDF（True）|13 MB本书介绍了拓扑学领域，这是一个探索几何空间性质的数学分支，重点是低维系统。作者讨论了在物理学各个领域的应用。本书的第一章涵盖了拓扑学的形式方面，包括类、同伦群、度量空间以及黎曼几何和伪黎曼几何。这些主题对于理解本书下一章中使用的理论概念和符号至关重要。这些应用包括晶体结构、空间拓扑、自旋统计、Braid群、Chern-Simons场论和3D引力等方面的缺陷。这本自足的书为物理和数学学生提供了所有必要的额外材料。演示文稿中充满了示例和练习，使读者能够轻松掌握概念。作者采用了一种教学方法，在简单的情况下提出了许多未解决的问题，这些问题可以作为学生具有挑战性的项目。适用于一学期的研究生课程，这篇文章非常适合教学目的。

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