Abstracts of the talks

Tania Bossio (fri 7 - 10:30)
Title: Tubular neighborhoods in sub-Riemannian geometry: Steiner's and Weyl's tube formulae

Abstract: Steiner and Weyl proved that the volume of the tubular neighborhood of a submanifold in Rn is a polynomial of degree n in the "size" of the tube. 

The coefficients of such a polynomial carry information about the curvature of the submanifold. 

In this talk, we investigate the validity of a Steiner and Weyl-like formula in the sub-Riemannian setting, extending previous results obtained when the ambient is a 3D contact sub-Riemannian space. 

We show that the volume of the tube around a non-characteristic submanifold of class $C^2$ is either smooth or real-analytic for small radii, whenever the ambient manifold and the assigned measure have both such a regularity.

This is a joint work with Luca Rizzi and Tommaso Rossi.

Nicola Garofalo (wed 3 - 14:00)

Title: Schrödinger semigroups and the Hörmander finite-rank condition"

Abstract: I introduce a class of (possibly) degenerate dispersive equations and prove that, under the H\"ormander finite-rank condition, the relevant Cauchy problem can be uniquely solved, and the solution operator can be uniquely extended to a strongly continuous semigroup in $L^2$. I will show that global solutions satisfy a sharp form of dispersive estimate in $L^p$ for which an uncertainty principle holds. This is joint work with Alessandra Lunardi.

Frédéric Jean (thu 4 - 11:30)
Title: Asymptotic geodesics and periodicity
Abstract: We consider the problem of characterizing sub-Riemannian geodesics that maximize asymptotically the average velocity, and we discuss whether the solutions correspond to periodic controls. This problem is motivated by application to locomotion modeling and by the study of periodic magnetic geodesics.

Antoine Julia (fri 5 - 14:00)
Title: Projections and (intrinsic) rectifiability.
Abstract: The Besicovitch projection theorem states that a $1$-dimensional set in the plane is rectifiable if and only if its orthogonal projection on lines have positive measure. It was generalized to every dimension by Federer. I will present the content of this theorem and whether we can hope to generalize it to the first Heisenberg group.

Alessia E. Kogoj (thu 4 - 14:00)
Title:  Asymptotic average solutions and a Pizzetti-type theorem for hypoelliptic PDEs

Abstract: By utilizing a Pizzetti's 1909 idea for the classical Laplacian, we introduced a notion of asymptotic average solutions. 

This notion enables the pointwise solvability of every Poisson equation $Lu(x)=−f(x)$ with continuous data $f$, where $L$  belongs to a class of hypoelliptic linear partial differential operators whose classical solutions can be characterized in terms of mean value formulae.

Antonio Lerario (wed 3 - 15:30)

Title: Sard properties for polynomial maps in infinite dimension

Abstract: In this talk I will consider "polynomial maps" from an infinite dimensional Hilbert space H to an m-dimensional Euclidean space E.

Here by "polynomial map" I mean a smooth map whose restriction to finite dimensional subspaces of H has components that are polynomials (of a fixed degree d).

Examples of maps belonging to this class are: (1) the map constructed by Kupka as a counterexample for the Sard theorem in infinite dimension; (2) Endpoint maps of Carnot groups. 

I will discuss some sharp results on the Sard property for this class of maps and then show how to apply these results in the sub-Riemannian context.

(For instance: I will show that on any Carnot group the Sard property holds true on the set of real-analytic controls with large enough radius of convergence.)

This is based on a joint work with Daniele Tiberio and Luca Rizzi.

Cyril Letrouit (thu 4 - 10:30)
Title: Generic controllability of equivariant systems and applications to particle systems and neural networks 

Abstract: There exist many examples of systems which have some symmetries, and which one may monitor with symmetry-preserving controls. Since symmetries are preserved along the evolution, full controllability is not possible, and controllability has to be considered inside sets of states with same symmetries. We prove that generic systems with symmetries are controllable in this sense. This result has several applications, for instance to particle systems, neural networks and eigenvalues of matrices. Joint work with Andrei Agrachev.

Paola Mannucci (fri 5 - 9:00)

Title: Non coercive first and second order evolutive Mean Field Games
Abstract: The Mean Field Games (MFGs) model describes interactions among a very large number of identical agents. Evolutive MFGs occur when the time horizon is finite; they are of first order if the dynamics of the agents are deterministic, of second order when the dynamics is stochastic.  They are modeled by a system of two coupled equations: a backward in time Hamilton- Jacobi equation and a forward in time continuity equation (Fokker-Planck for the second order case) describing respectively the optimal cost of a generic agent and the distribution of the whole population.
I will talk about some models of MFGs where the Hamiltonian is not coercive in the gradient term because the dynamics of the generic player must fulfill some constraints and fails to be strongly controllable.
First of all, I will outline the deterministic model where the generic player can move in the whole space, but it has some forbidden directions. I will show, as example, the Heisenberg case, both in periodic and in nonperiodic settings. We study the existence of weak solutions and we relate them with relaxed equilibria in the Lagrangian setting which describe the game in terms of probability measure on the set of optimal trajectories of the associated control problem.
The second part of the talk will be devoted to study subelliptic second order MFGs in Lie groups. In order to study the existence of classical solutions we first obtain new results about the nonlinear subelliptic Hamilton-Jacobi equation and separately the subelliptic Fokker-Planck equation.
References:
P. Mannucci, C. Marchi, N.Tchou "Non coercive unbounded first order mean field games: the Heisenberg example". J. Differential Equations (2022)
A. Cutri', P. Mannucci, C. Marchi, N.Tchou "The continuity equation in the Heisenberg periodic case: a representation formula and a application to Mean Field Games". NoDEA (2024)
P. Mannucci, C. Marchi, C. Mendico "Semi-linear parabolic equations on homogeneous Lie groups arising from mean field games" Math. Ann. (2024)

Julian Pozuelo (fri 5 - 11:30)

Title: Graphs of prescribed mean curvature in Heisenberg groups

Abstract: This talk is devoted to the study of the family of PDEs modeling those vertical graphs in Riemannian and sub-Riemannian Heisenberg groups with prescribed mean curvature without imposing Dirichlet conditions. Following results due to Giusti, we will prove existence of locally bounded $BV$ solutions in the so-called non-extremal domains. By an approximation procedure of the domain and the Riemannian space to the sub-Riemannian, we obtain locally bounded $BV$ solutions in extremal and non-extremal domains in both the Riemannian and sub-Riemannian settings. Then the regularity in the Riemannian setting will be improved in consecutive steps, obtaining a characterization of the existence of classical solutions of the Riemannian equation. These results are obtained in collaboration with Simone Verzellesi.

Tommaso Rossi (thu 4 - 15:00)
Title: The measure contraction property and the curvature-dimension condition in the sub-Finsler Heisenberg group
Abstract: The curvature-dimension condition, CD(K,N) for short, and the (weaker) measure contraction property, or MCP(K,N), are two synthetic notions for a metric measure space to have Ricci curvature bounded from below by K and dimension bounded from above by N. In this talk, we discuss the validity of these conditions in the sub-Finsler Heisenberg group. Firstly, we show that the CD(K,N) condition can not hold for any reference norm. Secondly, we show that the MCP(K,N) may hold or fail depending on the regularity of the reference norm. Moreover, when MCP(K,N) holds, we obtain estimates on the curvature exponent. This is a joint work with S. Borza, M. Magnabosco and K. Tashiro. 

Emmanuel Trélat (thu 4 - 9:00)
Title: From gas giant planets to the spectral theory of sub-elliptic Laplacians

Abstract: This is a work in progress with Yves Colin de Verdière, Charlotte Dietze and Maarten De Hoop, motivated by recent works by M. De Hoop on inverse problems for sound wave propagation in gas giant planets. 

On such planets, the speed of sound is isotropic and tends to zero at the surface. Geometrically, this corresponds to a Riemannian variety with an boundary whose metric blows up near the boundary. 

With appropriate variable changes, we can reduce the study of the Laplacian-Beltrami to that of a kind of sub-Riemannian Laplacian. 

In this talk, I'll explain how to approach the spectral analysis of such operators, and in particular how to calculate Weyl's law. 

Francesca Tripaldi (wed 3 - 16:30)

Title: Extracting subcomplexes on filtered manifolds

Abstract: I will present a general construction of subcomplexes on filtered manifolds. 

In the particular case of regular subRiemannian manifolds, this yields the so-called Rumin complex when the manifold is also equipped with a compatible Riemannian metric. 

I will then show how the subcomplex differs on a nilpotent Lie group equipped with a homogeneous structure on one hand, and a left-invariant filtration on the other.