This book offers a comprehensive and self-contained introduction to the geometric and algebraic foundations essential for modern theoretical physics, engineering, and materials science. It bridges two powerful mathematical frameworks—differential geometry and group theory—through a unified perspective centered on spatial transformations and coordinate invariance.

The first part develops the tools of differential geometry from the ground up, starting with tensors and their transformation properties in Cartesian and Minkowski spaces, progressing to differential forms, affine connections, and covariant derivatives on curved manifolds. These concepts culminate in the derivation of the geodesic equation and Einstein’s field equations, providing a rigorous pathway to general relativity via the Einstein-Hilbert action.

The second part introduces the structure and representation of groups, beginning with the axioms of group theory and advancing through discrete and continuous groups, Lie algebras, and representation theory. Special emphasis is placed on physical applications, including quantum mechanics, condensed matter systems, and symmetry principles underlying fundamental laws. Worked examples and character tables are presented in detail, alongside discussions of rotation groups (SO(2), SO(3)), SU(2), and the Lorentz and Poincaré groups.

Designed for graduate students, researchers, and final-year undergraduates, this textbook combines step-by-step derivations, intuitive explanations, and real-world applications, making it an indispensable resource for mastering the mathematical language of modern physics.

Alessio Zaccone is Full Professor of Physics at the University of Milan. His research spans quantum condensed matter physics, statistical mechanics, soft matter, and nuclear and high-energy physics, with a focus on developing theoretical frameworks that connect microscopic structure to macroscopic physical behavior. He previously held academic positions at the University of Cambridge and the Technical University of Munich, and was awarded the Gauss Professorship at the University of Göttingen. He is the recipient of a European Research Council (ERC) Consolidator Grant and has authored nearly 200 peer-reviewed publications in leading journals, as well as the Springer monograph Theory of Disordered Solids (2023). His interdisciplinary work bridges mathematics and physics across a wide range of modern theoretical research areas.