Topology and Dynamics of Higher-Order Networks - Lecture 1: General Introduction to Algebraic Topology on Networks and Simplicial Complexes

Explore the foundations of higher-order networks in this introductory lecture from the CMSP series on "Topology and dynamics of higher-order networks." Delve into the general introduction to algebraic topology on networks and simplicial complexes, focusing on homology and boundary operators. Learn how higher-order networks capture many-body interactions in complex systems and revolutionize our understanding of the interplay between topology and dynamics. Discover the emerging field of topological signals and its potential to transform our comprehension of structure-dynamics relationships in complex interacting systems. Examine how this field combines higher-order structures with discrete topology and dynamics, revealing new dynamical states and collective phenomena. Explore topological signals as dynamical variables sustained on nodes, edges, triangles, and higher-order cells of networks. Gain insights into the application of algebraic topology operators like the Hodge Laplacian and discrete Dirac operator in studying these signals. Investigate the collective phenomena exhibited by topological signals and their role in understanding how topology shapes dynamics and how dynamics learns network topology. Discover applications in mathematical physics and dynamical systems, and prepare for an in-depth exploration of this cutting-edge field over the course of four lectures and one seminar.