Explore cellular sheaves as a generalized concept of algebraic object networks in this comprehensive lecture. Begin with a simple introduction to the topic, then delve into the geometry-based definition of Laplacians for sheaves. Examine the Hodge theory that connects Laplacian geometry to sheaf algebraic topology. Learn how using the sheaf Laplacian as a diffusion operator enables sheaf dynamics, leading to decentralized methods for computing sheaf cohomology. Ground your understanding in real-world applications, focusing on social networks and opinion dynamics, with particular emphasis on consensus and polarization issues. Discover how these concepts apply to problems in community cleavage, personal opinion distributions, and online opinion dynamics. Investigate the use of lattices, Tarski chromology, and one-dimensional base spaces in realistic modeling. Gain insights from joint works with Jakob Hansen and Hans Riess in this talk, part of the "Topological Data Analysis - Theory and Applications" workshop. Read more