Assistant Professor Donald Sheehy was recently awarded two NSF/CISE/Directorate Grants for his research entitled, “CRII: AF: Principled Divide-and-Conquer for Topological Algorithms” and “AF:SMALL: Homological Methods for Big Enough Data”. These grants address fundamental questions in the growing field of Topological Data Analysis (TDA).
Topology is the mathematical study of shape. In many data sets, the data either describes something spatial or the ensemble of data points themselves have interesting shape such as clusters or loops. TDA tries to extract fundamental shape information from data sets while making a minimal set of assumptions about how the data was collected. This shape information is described as an invariant because it is unchanged by certain transformations of the underlying space. It is a growing field with applications in image analysis, biology, material science, as well as a host of other fields.
Two of the most fundamental problems in TDA are:
1) Efficiently computing invariants
2) Proving meaningful, general guarantees of correctness
The CRII grant addresses exclusively the first problem and is about bridging the gap from sparse matrix algorithms like nested dissection to TDA problems like persistent homology computation.
The AF: SMALL grant addresses the second problem and is about sampling theories and guarantees for data coverage by generalizing the theory of homological sensor networks.