Mircea Petrache (Pontificia Universidad Catolica, Santiago de Chile) will speak in the geometry seminar on Friday 28 February 2020 at 11h am in the Salle N 08.08 (this is the room behind the elevators in the 8th floor of the NO building). Mircea’s title is Uniform measures and manifolds all of whose curvatures are constant and his abstract is below.
A uniform measure in Euclidean space R^d is a measure with respect to which balls B(x,r) with center x in the support, are assigned mass dependent of r and independent of the choice x. For example any invariant measure with respect to a subgroup of the isometry group of R^d is uniform, and called a homogeneous measure. However we also have a few exotic examples of non-homogeneous uniform measures, such as the volume measure of the “light cone” {x^2+y^2+z^2=w^2} in R^4.This class of measures was first studied by David Preiss as the crucial ingredient of his 1987 proof of the Besicovitch conjecture. The complete classification of uniform measures remains a difficult open problem, even restricted to ambient dimension d=2. I will detail the known classification of 1-dimensional uniform measures in R^d for general d, for which, in joint work with Paul Laurain, we show that they are constituted of disjoint unions of helices or of toric knots, or equivalently, of analytic curves all of whose curvatures are constant.