Topological data analysis is based on an equally simple as intriguing principle. Leverage invariants from algebraic topology to gain novel insights into data. TDA is now applied in a wide variety of disciplines.

What is the probability that a random geometric graph in a sampling window has atypically few or atypically many edges or triangles? What are the sources leading to such rare events? These examples illustrate the core questions of large devations in geometric probability.

The concepts of percolation and chemical distances concern some of the most fundamental properties of large networks. Loosely speaking, they help to form an intuition what proportion of network nodes are connected and how long it takes of traveling from one node to another. While for simple combinatorial networks, these notions have been studied vigorously, for networks in Euclidean space with complicated spatial correlation patterns, the research is still in its infancy.