Definition of a manifold, weighted cohomology and metric matrices
The definition of the discrete Dirac operator, its properties and the discrete Dirac equation.

Prerequisites: Lecture 1. No other prerequisites other than basic notions of linear algebra are assumed.

Bianconi, G., 2021. The topological Dirac equation of networks and simplicial complexes. Journal of Physics: Complexity, 2(3), p.035022. Lloyd S, Garnerone S, Zanardi P. Quantum algorithms for topological and geometric analysis of data. Nature Communications. 2016 Jan 25;7(1):10138. Post, O., 2009, August. First order approach and index theorems for discrete and metric graphs. In Annales Henri Poincaré (Vol. 10, No. 5, pp. 823-866). Basel: SP Birkhäuser Verlag Basel.

Higher-order networks [1] capture the many-body interactions present in complex systems and are dramatically changing our understanding of the interplay between topology of and dynamics.
In this context, the new field of topological signals is emerging with the potential to significantly transform our understanding of the interplay between the structure and the dynamics in complex interacting systems.

This field combines higher-order structures with discrete topology, discrete topology and dynamics and shows the emergence of new dynamical states and collective phenomena.

Topological signals are dynamical variables, not only sustained on the nodes but also on edges, or even triangles and higher-order cells of higher-order networks. While traditionally network dynamics is studied by focusing only on dynamical variables associated to the nodes of simple and higher-order networks topological signals greatly enrich our understanding of dynamics in discrete topologies. These topological signals are treated by using algebraic topology operators as the Hodge Laplacian and the discrete Dirac operator.

Recently, growing attention has been devoted to the study of topological signals showing that topological signals undergo collective phenomena and that they offer new paradigms to understand on one side how topology shape dynamics and on the other side how dynamics learns the underlying network topology. These concepts and idea have wide applications. Here we cover example of their applications in mathematical physics, in dynamical systems and in topological machine learning.

The field is topical at the moment with many new results already established and an
already rich bibliography, therefore it is very timely to propose a series of lectures on the topic to introduce new scientists to this emergent field.

Here we propose a series of lectures for a broad audience of scientists addressed mostly to physicist and mathematicians, but including also computer scientists and neuroscientists.

The course is planned to be introductory, and self-contained starting from minimum set of prerequisites:

We will aim at producing some extensive lecture notes either before the lectures or soon after.

The course will cover 4 lectures and 1 seminar.