Hypergraphs provide a powerful mathematical framework for representing complex systems in which interactions occur not just between pairs of entities, but among groups. This advanced representation is particularly valuable for modeling diffusion, the spread of information or behaviors, through complex systems. Diffusion models play a key role in diverse fields such as biology, social science, and finance. On the other hand, diffusion on networks is a complex process where, for example, user characteristics may have a strong effect on how they shape and share their opinion. These factors play crucial roles in shaping diffusion patterns and need to be considered in modeling. The goal of this proposal is to design and explore a new mathematical tool, called the sheaf Laplacian, to better capture diffusion dynamics in hypergraphs by incorporating different factors like individual opinions, communication styles, and group roles. This will enable more flexible and accurate analysis of group interactions, with broad applications in areas such as expert detection, community decision-making, and collaborative information flow. This project develops and applies the theory of sheaf Laplacians on hypergraphs to address critical challenges in diffusion modeling, influence maximization, and representation learning. The investigators will first define generalized sheaf Laplacians for non-uniform hypergraphs and establish their key properties, including spectral characteristics, harmonic