Speakers

Title: On some variational problems involving functions with bounded Hessian

Abstract: Motivated by problems related to deep learning, we study variational problems involving bounded Hessian functions, i.e. functions whose gradient is a function of bounded variation. In this context, particularly interesting are density results for piecewise affine functions and questions regarding extremal points. Work in progress with M.Unser, S.Aziznejad, C.Brena.

Title: Connections between Heisenberg geometry and planar incidence geometry

Abstract: Projections along left cosets of a fixed horizontal line in the Heisenberg group relate aspects of Heisenberg geometry to incidence geometry in R^2. I will illustrate this phenomenon by means of some examples, including Loomis-Whitney inequalities in Heisenberg groups.

Title: Nilpotent groups, embeddings into L^1, and sets of finite perimeter in Carnot groups

Abstract: We shall present which are the nilpotent groups that admit a quasi-isometric embedding in the Banach space L^1 of integrable functions. We may consider finitely generated nilpotent groups equipped with word distances or nilpotent Lie groups equipped with left-invariant Riemannian metrics. From an asymptotic-cone argument we shall reduce to the case of bi-Lipschitz embeddings of Carnot groups. We shall prove that the only Carnot groups that embed are the abelian ones. From the work of Cheeger and Kleiner we shall see that for every Lipschitz map into L^1 one has a pullback distance obtained as a superposition of elementary distances with respect to cuts. Moreover, one only needs to consider cuts that have finite sub-Riemannian perimeter. The final goal is reached via a study of finite-perimeter sets and their blowups.

From a collaboration with S. Eriksson-Bique, C. Gartland, L. Naples and S. Nicolussi-Golo.

Title: Parabolic rectifiability

Abstract: I shall discuss in the n-space with parabolic norm definitions of rectifiability and characterizations in terms of approximate tangent planes and tangent measures. The results are analogous to those we proved with Raul and Franceso in 2010.

Title: A unified approach to extremal curves on Stiefel manifolds.

Abstract: We present a unified framework for studying extremal curves on real Stiefel manifolds. We start with a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. We find Euler-Langrange equations for a class of extremal curves that includes geodesics with respect to different Riemannian metrics and smooth curves of constant geodesic curvature. For some specific values of the parameter in the family of pseudo-Riemannian metrics we recover certain well-known used in the applied mathematics.

This is a joint work with K. Hueper (University of Wurzburg) and F. Silva Leite (University of Coimbra)

Title: Monotone sets and local minimizers for the perimeter in Carnot groups

Abstract: Monotone sets have been introduced about ten years ago by Cheeger and Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg groupinto $L^1$ to the classification of its monotone subsets. Later on, monotone sets played an important role in several works related to geometric measure theory issues in the Heisenberg setting. In this talk, we will present the notion of monotone sets in the wider setting of arbitrary Carnot groups: these are measurable sets for whichfor a.e. horizontal line $L$ the restriction of their characteristic function to $L$ agrees a.e. with a monotone function. We will explain recent results about measure-theoretic and topological properties of such sets, some of which require a detour through the study of local minimizers for the perimeter.

Title: Lipschitz functions on submanifolds of Heisenberg groups

We will present an almost everywhere "tangential" Pansu differentiability result for Lipschitz functions defined on intrinsically regular submanifolds of Heisenberg groups. This allows for a couple of applications, namely: a Lusin-type approximation for Lipschitz functions on intrinsic rectifiable sets, and a coarea formula on rectifiable sets. This is based on a joint work with A. Julia and S. Nicolussi Golo.