Geometry, Shapes & PDEs


Dr. Martin Bauer, Florida State University
Geometry, Shapes & PDEs

Thursday, March 31, 11 :00AM
In Person Event: 121 Manchester Hall

The past decades have seen tremendous advances in imaging techniques,
which have led to a significant growth in the quantity and complexity of data in
fields such as biomedical imaging, neuroscience and medicine.
Naturally, this prompted the emergence of new mathematical and algorithmic
approaches for the analysis of such data, which led to the emergence and
growth of fields such as geometric shape analysis and topological data analysis.
Infinite dimensional Riemannian geometry has proven to be a powerful tool to
deal with the challenges that arise in this context. In my talk I will give a short
introduction to the general concept of infinite dimensional Riemannian geometry,
where I will discuss several of the striking phenomena that might arise in this
situation. I will then focus on reparametrization invariant structures on spaces of
immersions and, in particular, I will introduce the class of Sobolev metrics on
spaces of curves and surfaces. For this class of Riemannian metrics I will discuss
the local and global well-posedness of the geodesic equations and properties of
the geodesic distance. Finally, to show how we can use this setup in practice, I
will discuss the numerical implementation of a statistical framework based on
such metrics.

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