New challenges across Analysis and Geometry

Eugenio Bellini
Classification of 3-dimensional K-contact structures

A contact sub-Riemannian manifold $M$ is called K-contact if the flow of the associated Reeb vector field acts on $M$ by sub-Riemannian isometries. K-contact manifolds are among the most important model spaces in sub-Riemannian geometry. Indeed, because of their symmetry, they provide an ideal framework for establishing connections between local sub-Riemannian invariants and global geometric properties of the ambient space. In this talk, I will present a complete classification of K-contact forms and K-contact structures in dimension three.

Samuel Borza
Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures

A classical result by Lott shows that a lower bound on the Bakry-Émery tensor in Riemannian geometry descends to suitable quotients. This result was later extended to the measure contraction property MCP(K, N) for essentially non-branching metric measure spaces, assuming that the quotient is by a compact group acting effectively via metric-measure isometries. I will explain how to replace the non-branching assumption with a new variant called δ-essential non-branching, and also relax the compactness condition to allow certain non-compact group actions. This is motivated by considerations in sub-Riemannian geometry, where δ-essential non-branching relates to a minimizing Sard property, and quotients of Carnot groups by (non-compact) stratified subgroups form the building blocks of sub-Riemannian structures. The new results I will present show that Carnot groups admitting a quotient to the Engel/Martinet structure cannot satisfy the MCP.

Tania Bossio
Magnetic fields on three-dimensional contact sub-Riemannian manifolds

The geodesic flow on the sub-Riemannian Heisenberg group can be interpreted as the lift of the magnetic geodesic flow of a charged particle moving in the Euclidean plane under a constant magnetic field.  Motivated by the classical correspondence between magnetic flows and sub-Riemannian geometry, first established by R. Montgomery, we investigate magnetic flows on three-dimensional contact sub-Riemannian manifolds. In this setting, magnetic fields are naturally described via Rumin differential forms, and the resulting magnetic geodesic flow corresponds to the geodesic flow of a lifted sub-Riemannian structure, which is of Engel type when the field is non-vanishing. In the general case, when the magnetic field may vanish, analytic properties of the magnetic field characterize geometric features of the lift, such as its step and abnormal curves.

Fabio Cavalletti
Timelike Ricci bounds in the non-smooth setting: an optimal transport approach

Optimal transport tools have been extremely powerful to study Ricci curvature, in particular Ricci lower bounds in the non-smooth setting of metric measure spaces (which can be been as a non-smooth extension of Riemannian manifolds). Since the geometric framework of general relativity is the one of Lorentzian manifolds (or space-times), and the Ricci curvature plays a prominent role in Einstein’s theory of gravity, a natural question is whether optimal transport tools can be useful also in this setting. The goal of the talk is to introduce the topic and to report on recent progress. More precisely: After recalling some basics of optimal transport, we will define "timelike Ricci curvature and dimension bounds" for a possibly non-smooth Lorentzian space in terms of displacement convexity of suitable entropy functions and discuss applications, including  the extension of classical singularity theorems to settings of low regularity, and prove a new  isoperimetric-type inequality.  Based on joint works with A. Mondino. 

Lorenzo Cecchi
Optimal transport between fibers

Antonini, Cavalletti and Lerario introduced, via a blend of symplectic geometry and optimal transport, a new structure on the space of projective hypersurfaces in CP^n. By mimicking their techniques, we introduce a geometric structure on the space of fibers of a holomorphic map. In search for insights, we will play with some examples in dimension one, which already exhibit interesting properties, such as compactness and geodesic convexity.
This is joint work with A. Lerario and F. Cavalletti.

Marco Di Marco
Stokes' Theorem in Heisenberg groups

We introduce the notion of submanifolds with boundary with intrinsic C^1 regularity in the setting of sub-Riemannian Heisenberg groups. We present a Stokes’ Theorem for such submanifolds involving the integration of Heisenberg differential forms introduced by Rumin. Talk based on a joint work with A. Julia, S. Nicolussi Golo and D. Vittone.

Karen Habermann
Geodesic and stochastic completeness for landmark space

In computational anatomy and, more generally, shape analysis, the Large Deformation Diffeomorphic Metric Mapping framework models shape variations as diffeomorphic deformations. An important shape space within this framework is the space consisting of shapes characterised by n ≥ 2 distinct landmark points in R^d. In diffeomorphic landmark matching, two landmark configurations are compared by solving an optimisation problem which minimises a suitable energy functional associated with flows of compactly supported diffeomorphisms transforming one landmark configuration into the other one. The landmark manifold Q of n distinct landmark points in R^d can be endowed with a Riemannian metric g such that the above optimisation problem is equivalent to the geodesic boundary value problem for g on Q. Despite its importance for modelling stochastic shape evolutions, no general result concerning long-time existence of Brownian motion on the Riemannian manifold (Q,g) is known. I will present joint work with Philipp Harms and Stefan Sommer on first progress in this direction which provides a full characterisation of long-time existence of Brownian motion for configurations of exactly two landmarks, governed by a radial kernel. I will further discuss joint work with Stephen C. Preston and Stefan Sommer which, for any number of landmarks in R^d and again with respect to a radial kernel, provides a sharp criterion guaranteeing geodesic completeness or geodesic incompleteness, respectively, of (Q,g).

Alessandro Tamai
Testing the algebraicity hypothesis

Given a probability measure on the unit disk, we study the problem of deciding whether, for some threshold probability, this measure is supported near a real algebraic set of bounded degree and given dimension. We call this ''testing the algebraicity hypothesis''. We prove an upper bound on the so called ''sample complexity'' of this problem and show how it can be reduced to a semialgebraic decision problem. This is done by studying in a quantitative way the Hausdorff geometry of the space of real algebraic sets of a given dimension and  degree."
This is a joint work with A. Lerario (SISSA), P. Roos Hoefgeest (KTH), M. Scolamiero (KTH)

Lucia Tessarolo
The Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds

Let $M$ be a 3-dimensional contact sub-Riemannian manifold and $S$ a surface embedded in $M$. Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation $\mathscr{F}$. In this talk we analyse the Schrödinger evolution of a particle constrained on $\mathscr{F}$.  Specifically, we define the Schrödinger operator $\Delta_\ell$ on each leaf $\ell$ as the classical "divergence of gradient", where the gradient is the Euclidean gradient along the leaf and the divergence is taken with respect to the surface measure inherited from the Popp volume, using the sub-Riemannian normal to the surface. We then study the self-adjointness of the operator $\Delta_\ell$ on each leaf by defining a notion of “essential self-adjointness at a point”, in such a way that $\Delta_\ell$ will be essentially self-adjoint on the whole leaf if and only if it is essentially self-adjoint at both its endpoints. We see how this local property at a characteristic point depends on a curvature-like invariant at that point. We then discuss self-adjoint extensions of an operator defined on the whole foliation.

Matteo Testa
Discriminants and Topological Complexity of Zero Sets on Manifolds

In numerical analysis, the complexity of a numerical problem is often quantified by its distance from the set of ill-posed problems, also known as the discriminant. Similarly, in real algebraic geometry, the distance of a polynomial from the discriminant has deep implications for the geometry of the associated variety and in many cases provides better informations than the ones given by the degree. In this talk we will explore how to define an analogous discriminant for the zero sets of smooth functions on a Riemannian manifold. The distance from this discriminant allows us to derive upper bounds on the Betti numbers of the zero set of such functions.
The talk is based on an ongoing joint work with Antonio Lerario and Saugata Basu.

Daniele Tiberio
Sard properties for polynomial maps in infinite dimension

Sard’s theorem asserts that the set of critical values of a smooth map from one Euclidean space to another one has measure zero. A version of this result for infinite-dimensional Banach manifolds was proven by Smale for maps with Fredholm differential. However, when the domain is infinite dimensional and the range is finite dimensional, the result is not true – even under the assumption that the map is “polynomial” – and a general theory is still lacking. In this seminar, I will provide sharp quantitative criteria for the validity of Sard’s theorem in this setting. As an application, I will present new results on the Sard conjecture in sub-Riemannian geometry. Based on a joint work with A. Lerario and L. Rizzi.