This book presents an overview of the mathematical foundations underlying modern theoretical physics, from Einstein's Relativity and the Standard Model to string theory and M-theory. It provides a gentle introduction to the algebraic and geometric language needed to understand these frameworks, focusing on curved spaces (Riemannian manifolds) and symmetries (Lie groups and their representations).
Key Features:
- Covers Lie theory and Riemannian geometry with applications to curvature decompositions, special holonomy, Einstein metrics, and M-theory
- Provides self-contained exposition of orthogonal groups, special subgroups, and their representations with geometric applications
- Accessible approach with proofs sometimes omitted or sketched to maintain focus on broader concepts
This work is aimed at postgraduate students and researchers interested in Riemannian geometry, mathematical physics, and Lie theory, with some material suitable for advanced undergraduates. It collects foundational material before progressing toward recent developments, making this rapidly evolving field more accessible.
Autorentext
Giovanni Russo studied mathematics at the Department of Mathematics, University of Turin (Italy), getting an interest for problems in analysis, differential geometry, and mathematical physics. In 2016 he moved to the Institute for Mathematics, Aarhus Universitet (Denmark), where he worked on nearly Kähler metrics with torus symmetry. There he got his Ph.D. degree in 2020. During that period, he visited the Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay, and Heriot-Watt University, Edinburgh.
Since 2020 he has been a postdoctoral researcher at various institutes. He visited the Max Planck Institute for Mathematics in Bonn (Germany), where he continued his research independently. Then he got postdoctoral fellowhips at the Philipps-Universität Marburg (Germany), at the University of L'Aquila (Italy), and at the Florida International University, Miami (United States). He is currently a postdoctoral researcher in differential geometry at the International School for Advanced Studies - SISSA, Trieste (Italy). He participated in many international conferences and gave a number of research talks in various countries. He also lectured for several courses at B.Sc., M.Sc., and Ph.D. level.
His areas of specialisation are mainly Riemannian geometry and Lie theory, with specific interests in Riemannian manifolds with special and exceptional holonomy, Einstein metrics, special geometries with symmetries, Lie theory, GKM theory of torus-actions, spin geometry, Morse theory, and topology of low dimensional manifolds.