The rationale for this workshop comes from recent developments bridging discrete and continuous worlds. On one hand, we have Gromov's work providing insight in approaching discrete metric spaces with geometric tools. On the other hand, the ideas born at the interface of topology and computer science (such as persistence) drastically improved our understanding of relations between the discrete samples from metric spaces endowed with structures, such as Riemannian manifolds, or semialgebraic sets.
The workshop aims to connect these two exciting areas using as testing ground the area where both approaches are highly relevant: the study of the structure of large networks. Networks, a catch-all meme in computer science, engineering, and the social sciences, are understood here rather formally as discrete simplicial complexes. Implicit or explicit enrichment of their structures - considering them as topological or metric spaces - often allows one to apply instruments from geometry and analysis normally not used to analyze social, biological, or engineered networks.
What are the new tools that geometry and topology provide? What is the structure of the spaces of networks? How to sample networks? These are the questions we plan to address, bearing in mind the numerous applications mentioned above.
The intended participants of the workshop are the experts in topology, geometry, and probability theory, as well as participants from statistics, computer science, biology, and engineering already working with large networks.