Proceedings of the International School of Physics "Enrico Fermi" - Course 215

This proceedings volume explores the interplay between topology and physics in the self-organization and emergent properties of condensed matter and materials, a topic that has gained major prominence in recent years. Across a wide range of systems, including entangled polymers with circular, linked, or knotted structures, liquid crystals and colloids, as well as topological insulators and Majorana fermions, topology plays a decisive role in generating novel physical phenomena and distinctive macroscopic material properties. Despite its central importance in contemporary condensed matter physics, topology is traditionally given limited attention in standard graduate curricula. The International School of Physics “Enrico Fermi” was conceived to address this gap by providing participants with a coherent and accessible introduction to both the essential mathematical concepts and the physical principles underlying current research at the interface of topology, hard matter, soft matter, and materials science. Based on lectures delivered by leading experts, this book offers authoritative and carefully structured lecture notes that bridge abstract topological ideas and concrete physical applications. Complemented by a critical summary from the editors, the volume presents a unified perspective on how topological concepts inform modern approaches to condensed matter and materials research. It is intended as a valuable resource for graduate students, researchers, and scientists seeking a deeper understanding of how topology shapes the behavior of complex physical systems.