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Understanding Random Systems via Algebraic Topology

Final Report Summary - URSAT (Understanding Random Systems via Algebraic Topology)

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, required the building of bridges between Topology and Probability, topics traditionally on opposite ends of Mathematics. This was at the core of URSAT. The next step was to use these bridges as connectors between Applied Topology and the methodology of Statistical Analysis.
Overall, URSAT not only met, but far exceeded, its original aims. In particular, a large team of postdoctoral fellows, with backgrounds from a broad range of disciplines (Topology, Probability, Statistics, Cosmology, and Electrical Engineering) worked together to create the synergy required by a project as broad in its aims as URSAT.
Results obtained led 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 URSAT.