Over the past decade there has been a significant activity in applying the techniques and theory of Algebraic Topology - perhaps the most esoteric branch of pure Mathematics - to real world problems. This expansion has generated new mathematical theory, new computational techniques, and even commercial startups. However, an important component of this topological approach that had not been treated in any depth previously was the inherently stochastic nature of the world and, in particular, the randomness inherent in any data collection.

URSAT, and the work leading up to it, were designed to overcome this deficiency. The first steps, on the theoretical side, require the building of bridges between Topology and Probability, topics traditionally on opposite ends of Mathematics. This is at the core of URSAT. The next step will be to use these bridges as connectors between Applied Topology and the methodology of Statistical Analysis. While this is part of the ultimate aim of URSAT, it is not a primary part of the program.

The past two and a half years have seen impressive progress towards fulfilling URSAT’s aims. In particular, a large team of postdoctoral fellows, with backgrounds from a broad range of disciplines (Topology, Probability, Statistics, Cosmology, and Electrical Engineering) are now working together to create the synergy required by a project as broad in its aims as URSAT.

Results to date have lead to a much deeper understanding than had existed previously regarding the topological nature of random structures (technically - simplicial complexes) built from point cloud data of many types, as well as of low dimensional random surfaces (manifolds), both in high dimensions. These results are helping to help provide theoretical insight behind the current methods of Topological Data Analysis, most of which are somewhat ad hoc, and, more importantly, to open the way for the development of additional methodologies for the analysis of what today is generically known as “Big Data”.

The message that Topology and Probability can not only be combined, but that there is a unifying thread here, is beginning to be acknowledged by the broader Mathematics and Statistics communities. The fact that it uncovers new Mathematics, as well as having practical importance for Data Analysis, is what is leading to this, and it is this acknowledgement, perhaps more than anything else, which is a measure of the success of the first two and a half years of URSAT.