Functoriality in Geometric Data

Summary

Across science, engineering, medicine and business we face a deluge of data coming from sensors, from simulations, or from the activities of myriads of individuals on the Internet. The data often has a geometric character, as is the case with 1D GPS traces, 2D images, 3D scans, and so on. Furthermore, the data sets we collect are frequently highly correlated, reflecting information about the same or similar entities in the world, or echoing semantically important repetitions/symmetries or hierarchical structures common to both man-made and natural objects.

A recent trend, emerging independently in multiple theoretical and applied communities is to understand geometric data sets through their relations and interconnections, a point of view that can be broadly described as exploiting the functoriality of data, which has a long tradition associated with it in mathematics. Functoriality, in its broadest form, is the notion that in dealing with any kind of mathematical object, it is at least as important to understand the transformations or symmetries possessed by the object or the family of objects to which it belongs, as it is to study the object itself. This general idea been successfully applied in a large variety of fields, both theoretical and practical, often leading to deep insights into the structure of various objects as well as to elegant and efficient methods in various application domains, including computational geometry, computer vision and computer graphics.

This seminar brought together researchers and practitioners interested in notions of similarity, correspondence and, more generally, relations across geometric data sets. Mathematical and computational tools for the construction, analysis, and exploitation of such relational networks were the central focus of this seminar.