Geometry and Topology of Low Dimensional Systems

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.

Offers a pedagogical approach to the interplay between topology and physics Discusses several applications in topological field theory, space topology and low dimensional geometry Enriched with exercises to master the subject

Autorentext

Ramadevi Pichai is a Professor in the Department of Physics at the Indian Institute of Technology Bombay, Mumbai. Her research focuses on mathematical physics, particularly in areas such as topological field theory, knot theory, string theory, and supersymmetry. She has dedicated her work to understanding Chern-Simons field theory as it relates to knots and links, exploring their connections to topological string theory. With a publication record exceeding 50 papers in international journals, she has successfully supervised six PhD students. Over the past 22 years, Professor Pichai has taught numerous undergraduate-level courses. She organized a national school at IIT Bombay and edited the lectures, which were published four years ago by Hindustan Book Agency. Recently, she coauthored an undergraduate-level textbook on "Group Theory for Physicists," published by Cambridge University Press.

T. R. Govindarajan has a distinguished career spanning over two decades at the Institute of Mathematical Sciences, Chennai. Formerly a Professor and Dean of Homi Bhabha National Institute, IMSc center, he has held positions such as Senior Associate of ASICTP and visiting researcher or professor at renowned institutions including Texas University at Austin, Syracuse University, University of Sao Paulo, Max Planck Inst. for Gravitation Physics, Dublin Institute for Advanced Studies, among others. With four students completing their PhD under his guidance, Govindarajan also teaches at the Chennai Mathematical Institute. His extensive publication record exceeds 100 publications, and he serves on the editorial teams of various journals. Govindarajan played a pivotal role in the Advisory group for Theoretical Physics at S N Bose Inst. He has organized numerous international conferences and actively participated in various committees.

Inhalt
Introduction.- Differentiable Manifolds and Geometry.- Riemannian and Pseudo Riemannian Geometry.- Topological Understanding of Defects in Crystalline Structure.- Configuration Space Topology and Topological Conservation Laws.- Spin Statistics Theorem, and Low Dimensional Geometry and Topology.- Braid Group, Knots, Three Manifolds.- Chern-Simons Field Theory.- 3D Gravity as Topological Quantum Field Theory.