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-theoryProvides self-contained exposition of orthogonal groups, special subgroups, and their representations with geometric applicationsAccessible approach with proofs sometimes omitted or sketched to maintain focus on broader conceptsThis 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.
