Mathematical Methods for Physics: An Introduction to Group Theory, Topology and Geometry

This detailed yet accessible text provides an essential introduction to the advanced mathematical methods at the core of theoretical physics. The book steadily develops the key concepts required for an understanding of symmetry principles and topological structures, such as group theory, differentiable manifolds, Riemannian geometry, and Lie algebras. Based on a course for senior undergraduate students of physics, it is written in a clear, pedagogical style and would also be valuable to students in other areas of science and engineering. The material has been subject to more than twenty years of feedback from students, ensuring that explanations and examples are lucid and considered, and numerous worked examples and exercises reinforce key concepts and further strengthen readers' understanding. This text unites a wide variety of important topics that are often scattered across different books, and provides a solid platform for more specialized study or research.

Table of Contents:
1. Introduction; 2. Group theory; 3. Representation theory of groups; 4. Differentiable manifolds; 5. Riemannian geometry; 6. Semisimple Lie algebras and their unitary representations; Appendix A; References; Index.

About the Author :
Esko Keski-Vakkuri received his Ph.D. in physics from Massachusetts Institute of Technology in 1995, and is currently a senior faculty member at the University of Helsinki. He has previously held positions at the California Institute of Technology and Uppsala University. His research is focused on string theory, black holes, holographic duality, and quantum information. Claus K. Montonen received his Ph.D. from the University of Cambridge in 1974 and later held positions at the Université de Paris XI (CNRS), the Research Institute for Theoretical Physics, Helsinki, and CERN. He held various senior faculty positions in the Department of Physics at the University of Helsinki from 1978 until his retirement in 2011, where he was responsible for curriculum design in theoretical physics. His research interests are in S-matrix theory, string theory and quantum field theory, having made major contributions to early string theory and duality in field and string theory. Marco Panero received his Ph.D. in physics from the University of Turin in 2003, after which he held postdoctoral positions at the Dublin Institute for Advanced Studies, the University of Regensburg, ETH Zurich, the University of Helsinki, and the Autonomous University of Madrid. Since 2014 he has been an Associate Professor in physics at the University of Turin. His main research interests are in lattice field theory and in theoretical high-energy physics.

Review :
'The recent explosive development of topological quantum matter requires a deep systematic understanding of modern mathematics. Quantum many-body entanglement in topological quantum matter is a new phenomenon that requires new mathematical language to describe. This is a rare book that provides systematic and in-depth coverage of some of the most important mathematical concepts, such as groups, geometry, topology and algebra, among others. Many abstract mathematical notions are explained in an easy, explicit fashion. This is an in-depth, friendly book on modern mathematics. Very timely and highly recommended.' Xiao-Gang Wen, Massachusetts Institute of Technology