AMS Seminar: Nicolas Charon (JHU- Applied Math & Stats) @ Whitehead 304

Title: “Measures in geometry: a look at two cases of fruitful interaction.”
Abstract: “This talk will explore the use of measures as a convenient way to represent and analyze the shape of objects, mathematically and/or numerically. We will specifically focus on two particular examples of such interactions.
In the first part of the talk, I will introduce, in a simple setting, the so called length measures associated to planar closed curves. Although the length measure do not fully characterize the underlying shape, the celebrated Minkowski-Fenchel-Jessen theorem shows that this is the case when restricting to the subclass of convex shapes. This equivalence has been key to the derivation of many important results in the field of convex geometry. We will mention some of these such as isoperimetric inequalities and discuss several open questions related to length measures for non-convex shapes.
In the second part of the talk, I will present another class of measures called varifolds, which allow this time to represent injectively any shape such as curves, surfaces or even submanifolds of any dimension. I will then examine the construction of numerically tractable metrics based on varifolds which can be used to formulate and tackle various problems in shape analysis. We will focus in particular on the problems of compression (or quantization) of varifolds and of diffeomorphic registration between two shapes.”