Abstract

In this seminar I present some ideas and recent results in the space of geometric modelling of complex adaptive systems, which are driven by applications on real-world networks and deep (Graph) NN architectures. Modelling complexity and emergent phenomena has been an active field of research in recent years. Relevant properties of real-world systems are often collective/emergent. Similarly, the learning process, behaviour, and key characteristics of deep NN architectures (also complex adaptive systems) are intrinsically emergent.

The geometric approach is directly related to a successful paradigm in modelling emergent phenomena. I introduce a specific construction of a principal bundle over simplicial complexes, which naturally arises from local multi-agent interactions in a system. This construction leads to consistent definitions of discrete analogues of classical obstructions to the integrability of geometric structures on manifolds (which are geometric such as curvature and torsion forms and topological i.e. captured by characteristic classes). These objects, ultimately related to the non-triviality of principal bundles, are the key elements of a consistent and rich gauge theory. Unsurprisingly these ideas have direct relation and potential implications for the more recently developed field of Geometric Deep Learning.