Given a real odd-dimensional oriented submanifold $M$ of a complex manifold $X$ a natural question arises: is $M$ the boundary of a complex submanifold (or subvariety) $Y$ of $X$? This problem, known as the boundary problem, is now more than 60 years old, but it is still alive, since the question can be asked in various different settings. Ideally the task would be to find necessary and sufficient conditions on $M$. The first answer was due to Wermer and dates back to 1958: the case of a compact curve in $C^n$ was solved. Almost two decades later (1975 and 1977), Harvey and Lawson dealt with the case of a compact manifold of arbitrary odd-dimension in $C^n$. In order to get a solution, the regularity request on $Y$ had to be dropped, allowing also holomorphic chains as weak solution (having $dY=M$ in the sense of currents). Adding some convexity properties on $M$ allowed to prove that the solution $Y$ is indeed a variety. More recently, in the '90s, the much more difficult problem for $M$ in some open subset of $CP^n$ was tackled by Dolbeault, Henkin and Dihn, and in the new millennium the problem in $CP^n$ was approached by Harvey, Lawson and Wermer again. The unbounded version of the problem (i.e. when $M\subset C^n$ is not compact) was studied by Della Sala and myself, and the problem in Hilbert complex space by Mongodi and myself. I will review known results and open problems.

At the beginning of the seminar, we introduce the setting of Carnot groups and, in this realm, we discuss the link between different notions of rectifiability: the general one for metric spaces by Federer and the ones proposed in the specific case of Carnot groups by Pauls and Franchi, Serapioni and Serra Cassano. We focus on codimension-one rectifiability and present some known results that highlight the relations between the various notions we present. In the second part of the seminar we present our results. First, we show the existence of a smooth non-characteristic hypersurface in a Carnot group of topological dimension 8 that is purely Pauls unrectifiable. This shows that even very regular objects of codimension one that for sure are rectifiable according to the definition of Franchi, Serapioni and Serra Cassano are not so according to Pauls’ definition. Finally we show that such an example does not exist in the Heisenberg group H^n, with n ≥ 2. In particular we show that every smooth hypersurface in H^n, with n ≥ 2, is Pauls rectifiable, even with bi-Lipschitz maps. For n = 1 this last result was already known from previous works by Cole, Pauls and Bigolin, Vittone. This is a joint work with Enrico Le Donne. The results we present are contained in the arXiv preprint "Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces" (https: //arxiv.org/abs/1910.12812).

In a joint work with Giovanna Citti (Bologna), and Nicola Garofalo (Padova) we establish gradient regularity for solutions of a class of degenerate parabolic PDE modeled on the regularized p-Laplacian (p larger or equal than 2) in the Heisenberg group or more in general on subriemannian contact manifolds . Our approach rests on a set of techniques introduced by X. Zhong in the stationary case, which we extend to the time-dependent PDE.

Between metric spaces, sublinear bilipschitz equivalences are generalized quasiisometries for which the large-scale Lipschitz behavior is kept, while the coarse behavior is not. They appear between families of (non-pairwise quasiisometric) nilpotent or solvable connected Lie groups with close structure, in the study of asymptotic cones of Lie groups by Cornulier. In this talk, I will report on my work on revisiting certain classical quasiisometric invariants of groups to determine which of them can be turned into sublinear asymptotic invariants, in the continuation of Gromov's questions of classifying homogeneous spaces (e.g. symmetric spaces, noncompact solvmanifolds) up to quasiisometry and investigate their quasiisometric rigidity. I will present a partial classification result for the Riemannian homogeneous spaces of negative curvature. Time permitting, I will discuss the (open) problem of determining which finitely generated groups are sublinearly bilipschitz equivalent to real hyperbolic spaces, and a quantitative approach to the classification on the nilpotent side.

We introduce the Moduli space of sub-Riemannian Heisenberg manifolds and show an analogy of Mahler's compactness theorem, that is, any subset with lower bound of systoles and upper bound of measure is compact.

Many homogeneous metric spaces are actually (solvable) Lie groups equipped with left-invariant distances. I will first show some results regarding how general is the the existence of such Lie group structures on homogeneous metric spaces. Then, I present a new method on how to find all the possible structures, given the existence. This method is the construction of so called real shadow of a solvable Lie group. It generalises the nilshadow of Auslander and Green, and Alexopoulos, and provides a unified framework to find completely solvable representatives for solvable groups. The existence of these representatives was earlier understood only for Heintze groups on the one hand and for the groups of polynomial growth on the other.

In this seminar, first I will recall some necessary material about the geometry of semi sprays, Lagrangian and Finsler geometries. Then I will show the meaning of "inverse problem of Lagrangian mechanics" which is called Finsler metrizabilty problem in the Finsler context along with some known results. After that going to the relation between Frobenius integrability of certain distribution and the metrizabilty problem. Finally, I will present my joined research work in this problem based on "I. Bucataru, G. Cretu and Ebtsam H. Taha, Fronenius integrability and Finsler metrizability for two-dimensional sprays, Diff. Geom. Appl., 56 (2018), 308-324". In more details, for a 2-dimensional non-flat spray we associate a Berwald frame and a 3-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterizes the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a 1-form from the annihilator of the Berwald distribution. Moreover, I will give some examples to show how our mechanism is working and it is a constructive one.

Orlicz cohomology is a generalization of Lp-cohomology, which is an important quasi-isometry invariant. It is possible to define it (as in the Lp-case) in a de Rham version and in a simplicial version (where it is not difficult to prove its quasi-isometry invariance). In the Lp-case it is known that both versions are, in some sense, equivalent. A work by Matías Carrasco proves the equivalence between both versions of Orlicz cohomology in degree one, and shows very interesting applications to the large scale geometry of Heintze groups. This motivates the study of Orlicz cohomology in higher degrees (in particular in the case of Heintze groups), for which it is important to have a more general equivalence theorem. The goal of the talk will be to present an equivalence theorem between Orlicz-de Rham cohomology and simplicial Orlicz cohomology in the case of Lie groups. This will allow to conclude the quasi-isometry invariance of Orlicz-de Rham cohomology in that case.

Motivated by Gromov’s precompactness theorem Lott, Villani and Sturm defined the notion of synthetic Ricci curvature bound on a metric measure space. This definition is formulated in terms of the convexity of an entropy functional along geodesics in the space of probability measures and is known as the Curvature-Dimension condition (CD(K,N)). Isometric actions on Riemannian manifolds have been a useful tool to investigate the interaction between the topology and the Riemannian metric a manifold might admit. A major result in this area is the theorem of Myers-Steenrod stating that the isometry group of a Riemannian manifold is a Lie group. In this talk I will look at the isometry group of an RCD*(K,N) space, prove that it is a Lie group and, if time permits, I will discuss what can be done to ensure that a compact Lie group acts by measure preserving isometries.

We will explore the way that Heintze groups, as one-dimensional extensions of nilpotent Lie groups, play the role of natural generalizations of rank one symmetric spaces. In doing so we observe that these classical examples have additional structure in the form of a Carnot grading on their derived subalgebra, a condition called Carnot-type. We construct a left-invariant metric on Heintze groups of Carnot-type which mimics the symmetric metrics on real, complex, and quaternionic hyperbolic space. Notably this metric produces pinched curvature which generalizes the quarter-pinching of the rank one symmetric spaces - we demonstrate the sharpness of this bound under a mild assumption.

Brascamp--Lieb inequalities are multilinear integral inequalities, classically set in Euclidean spaces, in which the functions involved possess symmetries that can be described via annihilation by translation invariant vector fields. This point of view can be translated into other non-Euclidean settings and adapted to establish similar inequalities. The main tool is a monotonicity property of the heat flow. In this talk I will explore two cases. The first is the compact case of real spheres, where the argument becomes ``discrete'' and has a combinatorial flavor and where the inequalities are seen to be sharp for certain choices of symmetries. The second setting is the non-compact case of stratified Lie groups, where thanks to the homogeneity properties of the heat kernel we can adapt the Euclidean techniques. As an application, we deduce a Loomis--Whitney type inequality for stratified groups and derive sub-Riemannian isoperimetric inequalities from it.

In an arbitrary Carnot groups, we consider non-local perimeters given by positive symmetric kernels with suitable integrability conditions. In the first part of the talk, we study the local minimizers for these perimeters in a fixed open set. By means of a notion of calibration introduced by V. Pagliari, we show that vertical halfspaces are unique minimizers for such non-local perimeters in the unit ball, when subject to their own outer datum. Using some tools introduced by J. Cabré, we show how slightly stronger assumptions can lead to uniqueness of minimizers in a generic bounded open set. In the second part of the talk, we discuss the Gamma-convergence of the rescaled non-local perimeters to a measure which is explicit and absolutely continuous with respect to the (local) sub-Riemannian perimeter. This is a joint work with Alessandro Carbotti, Diego Pallara and Andrea Pinamonti

In this talk I will consider the classical problem of rectifiability of boundaries of finite-perimeter sets in Carnot groups. In Euclidean spaces rectifiability can be equivalently described in terms of $C^1$-hypersurfaces, Lipschitz-graphs, or a geometric condition which we call a "cone property". In Carnot groups these different types of rectifiability have their natural counterparts, but their equivalence is still unknown. I will describe results regarding all of the rectifiability types mentioned above and give examples of new classes of Carnot groups where a rectifiability result is obtained. This talk is based on joint work with Sebastiano Don, Enrico Le Donne and Davide Vittone.