open problems

Monday, May 21, 2018 - 10:15am - 11:15am
Elizabeth Munch (Michigan State University)
Reeb graphs and mapper graphs are tools from Applied Topology and TDA which skeletonize and summarize the shape and structure of data. In this open-directions talk, we discuss how one can view these graphs as sheaves. This particular viewpoint leaves open the possibility of using ideas from the interleaving distance for persistence modules to create a metric for the structures. We will discuss various theorems showing when the mapper graph converges to the Reeb graph and discuss open questions related to these ideas to seed discussion during the workshop.
Monday, May 21, 2018 - 3:15pm - 4:15pm
Emilie Purvine (Pacific Northwest National Laboratory)
When using sheaves to model real-world problems -- e.g., topological signal processing or information integration -- we must face up to the challenges that real data provide. In a sheaf-theoretical context, measured data, called assignments, are represented by members of objects in the data category. Assignments need not be sections, and indeed, as measured data, almost always require statistical descriptions and tolerances. Uncertainty can exist between disparate measurements generally, and even on the same observable, due to uncertainty, noise, or sensor malfunctions.
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