PaPa

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Monday 6

9h30 Registration and Welcome

10h Hajer Bahouri: What about the dispersive phenomenon  for evolution equations on the Engel group

Abstract: In this talk, we will  emphasize the influence of the geometry of the group and particularly the structure of the sub-laplacian on dispersive phenomena. While the Schrödinger equation on the Heisenberg behaves as a transport equation with respect to the center variable, it turns out that on the Engel group, the Schrödinger equation has  the following form: $$ i\partial_t - |D_{s}|^{2/3} \cH (D),$$

where $s$ is the center variable and $\cH (D)$ is a   one dimensional pseudo-differential operator of order $0$. It is a joint work in progress with Davide Barilari, Isabelle Gallagher and Matthieu Léautaud.

11h Luca Rizzi: A unifying theory of Ricci curvature lower bounds for sub-Riemannian manifolds

Abstract: We report on an on-going work in collaboration with Barilari (Padova) and Mondino (Oxford) concerning a general theory of curvature-dimension bounds for metric measure spaces that includes smooth and non-smooth sub-Riemannian structures. Our theory contains as a special case the Lott-Sturm-Villani's one, and can be thus seen as a step towards Villani's ``great unification'' program of curvature-dimension bounds of Riemanniann, Finslerian, and sub-Riemannian geometries.

14h Alessio Martini: Spectral multipliers for sub-Laplacians: recent developments and open problems

Abstract: I will present some old and new results about the $L^p$ functional calculus for sub-Laplacians $L$.

It has been known for a long time that, under fairly general assumptions on the sub-Laplacian and the underlying sub-Riemannian structure, an operator of the form $F(L)$ is bounded on $L^p$ ($1<p<\infty$) whenever the multiplier $F$ satisfies a scale-invariant smoothness condition of sufficiently larger order.

The problem of determining the minimal smoothness assumptions, however, remains widely open and will be the focus of our discussion.

15h Alessandro Socionovo: Non-minimality of sub-Riemannian spirals

Abstract: The authors show that in analytic sub-Riemannian manifolds of rank 2 satisfy ing a commutativity condition, spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.

Tuesday 7

9h30 Annalisa Massaccesi: Frobenius theorem for weak submanifolds

Abstract: The question of producing a foliation of the n-dimensional Euclidean space with k-dimensional submanifolds which are tangent to a prescribed k-dimensional simple vectorfield is part of the celebrated Frobenius theorem: a decomposition in smooth submanifolds tangent to a given vectorfield is feasible (and then the vectorfield itself is said to be integrable) if and only if the vectorfield is involutive. In this seminar I will summarize the results obtained in collaboration with G. Alberti, A. Merlo and E. Stepanov when the smooth submanifolds are replaced by weaker objects, such as integral or normal currents or even contact sets with "some" boundary regularity. I will also provide Lusin-type counterexamples to the Frobenius property for rectifiable currents. Finally, I will try to highlight the connection between involutivity/integrability à la Frobenius and Carnot-Carathéodory spaces and how to apply our techniques in this framework. 

10h30 Daniele Cannarsa: Induced geometry on surfaces in 3D contact sub-Riemannian manifolds

Abstract: Any subspace S of a metric space M admits an induced (possible infinite) length distance, by defining the distance between two points in S as the infimum of the lengths of the curves in S connecting the points. In this talk we discuss what happens in the setting of a surface S embedded in a 3D contact sub-Riemannian manifold M. In particular, we identify some global conditions for the induced distance to be finite, and we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.  [Joint work with Davide Barilari and Ugo Boscain.]

12h  Ugo Boscain: Embedding of the Grushin cylinder in R^3 and its geometric quantization

Abstract: In this talk I am going to discuss how to write the Schroedinger equation for the Grushin cylinder. I will discuss intrinsic (geometric) quantizations and extrinsic quantizations via an embedding in R^3. The main problem being to understand if a quantum particle can cross the singular set. Generalizations to almost-Riemannian structures will be discussed.