"Shape Analysis: The challenge of geometric data"
, 499 DSL
Despite tremendous volume and rapidity of data being collected in different applications, it is a common belief that the underlying structures are typically low-dimensional albeit non-Euclidean. The use of differential geometry in extracting such low-dimensional structures and imposing probabilistic superstructures on these extractions is a common way of analyzing data nowadays. There is a subclass of problems where the extracted structures are restricted to a certain topology and the focus is on analyzing variability in the geometries of the extracted structures. One such field of study is statistical shape analysis in which one tries to quantify geometric variability within and across populations using statistical tools. This has been the focus of intense research in recent years. In my talk I will describe several challenges that arise in applications, where each data point is itself a (complex) geometric object. In particular, I will consider collections of shapes that consist of curves, surfaces or even higher dimensional manifolds. I will show how infinite dimensional geometry provides a natural and convenient setup and will present both the mathematical challenges and the obtained numerical frameworks.