Talks and abstracts
Future seminars
26/06/2020 - Luca Rizzi (Grenoble)
Title: Heat content asymptotics for sub-Riemannian manifolds [SLIDES]
Abstract: We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series for small times, at arbitrary order. We compute explicitly the coefficients up to order k = 5, in terms of sub-Riemannian invariants of the domain. Furthermore, as an independent result, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a particular case we recover, using non-probabilistic techniques, the order 2 formula recently obtained by Tyson and Wang in the Heisenberg group [Comm. PDE, 2018]. A consequence of our fifth-order analysis is the evidence for new phenomena in presence of characteristic points. In particular, we prove that the higher order coefficients in the asymptotics can blow-up in their presence.
This is a joint work with T. Rossi (Institut Fourier & SISSA)
Past seminars
17/04/2020 - Enrico Le Donne (Pisa and Jyväskylä)
Title: Mathematical appearances of sub-Riemannian geometries [SLIDES]
Abstract: Sub-Riemannian geometries are a generalization of Riemannian geometries. Roughly speaking, in order to measure distances in a sub-Riemannian manifold, one is allowed to only measure distances along curves that are tangent to some subspace of the tangent space. These geometries arise in many areas of pure and applied mathematics (such as algebra, geometry, analysis, mechanics, control theory, mathematical physics, theoretical computer science), as well as in applications (e.g., robotics, vision). This talk introduces sub-Riemannian geometry from the metric viewpoint and focus on a few classical situations in pure mathematics where sub-Riemannian geometries appear. For example, we shall discuss boundaries of rank-one symmetric spaces and asymptotic cones of nilpotent groups. The goal is to present several metric characterizations of sub-Riemannian geometries so to give an explanation of their natural manifestation. We first give a characterization of Carnot groups, which are very special sub-Riemannian geometries. We extend the result to self-similar metric Lie groups (in collaboration with Cowling, Kivioja, Nicolussi Golo, and Ottazzi). We then present some recent results characterizing boundaries of rank-one symmetric spaces (in collaboration with Freeman).
15/05/2020 - Maria Karmanova (Moscow)
Title: A New Look at Carnot-Caratheodory Spaces Theory and Related Topics [SLIDES]
Abstract: In my talk, I'll present results on fine local metric properties of Carnot-Caratheodory spaces under minimal assumptions on smoothness of basis vector fields. These properties are new even for the classical "smooth" case. Moreover, they can be considered as new foundations of the theory. I'll also review main corollaries, including non-holonomic geometric measure theory formulas for wide classes of mappings in sub-Riemannian and sub-Lorentzian geometry, and some applications for minimal and maximal surfaces theory.
12/06/2020 - Tuomas Orponen (Helsinki)
Title: Sub-elliptic boundary value problems in flag domains [SLIDES]
Abstract: I will talk about solving the sub-Laplacian Dirichlet and Neumann problems with L2 boundary data in “flag domains” of the first Heisenberg group. These are domains bounded by a vertically ruled Lipschitz graph. The solutions are obtained via the method of layer potentials. This is joint work with Michele Villa.