I will introduce a class of spaces that can be seen as an infinite-dimensional analogue for Carnot groups. These spaces cover classical, finite-dimensional Carnot groups as well as separable Banach spaces. I will also present a version of Rademacher's theorem for Lipschitz functions with infinite-dimensional Carnot group domains and give examples of noncommuting infinite-dimensional spaces in which our theorem can be applied. This is joint work with Enrico Le Donne and Sean Li.

We give a characterization of BV functions in metric measure spaces making use of suitable vector fields. This constitutes the starting point for a discussion on Gauss-Green Formulæ featuring the normal traces of divergence-measure vector fields, and for new results on the traces of BV functions. Based on joint works with G. E. Comi and M. Miranda Jr.

By definition, the $\ell^{q,p}$ cohomology of a bounded geometry Riemannian manifold is the $\ell^{q,p}$ cohomology of every bounded geometry simplicial complex quasiisometric to it. Following a result by Pansu and Rumin, we know that every $\ell^{q,p}$ cohomology class of a contractible Lie group can be represented by a form $\omega$ which belongs to $L^p$ as well as an arbitrary finite number of its derivatives. If the class vanishes, then there exists a primitive $\phi$ of $\omega$ which belongs to $L^q$ as well as an arbitrary finite number of its derivatives. This holds for all $1\leq p\leq q\leq\infty$. Recent results in collaboration with Pansu show the sharp range of values of $q$ integer for which the $\ell^{q,1}$-cohomology of the Heisenberg groups vanishes.

The regularity of length-minimizing curves on sub-Riemannian manifolds is a mystery. In this talk, I will discuss the metric blowup approach to the problem. The key ingredients are the iterative study of tangent cones of geodesics, and the cut-and-correct method to constructing competitors to potential length-minimizers. This talk is based on joint work with Enrico Le Donne.

The $l_\infty$ left invariant problem on Cartan group $M$ is studied as the following time-optimal problem for $q = (x,y,z,v,w) \in \mathbb{R}^5 = M$: \begin{equation} \dot{q} = u_1 X_1 + u_2 X_2, \qquad \max (|u_1|, |u_2|) \leq 1, \quad (1) \end{equation} \begin{equation} q (0) = q_0 = (0, \dots, 0), \qquad q(T) = q_1, \quad (2) \end{equation} \begin{equation}T \to \min, \quad (3) \end{equation} where $X_1 = \frac{\partial}{\partial x} - \frac{y}{2} \frac{\partial}{\partial z} - \frac{x^2 +y^2}{2} \frac{\partial}{\partial w}$, $X_2 = \frac{\partial}{\partial y} + \frac{x}{2} \frac{\partial}{\partial z} + \frac{x^2+y^2}{2} \frac{\partial}{\partial v}$. Pontryagin maximum principle is applied to problem (1)-(3). The normal case provides the following types of extremal arcs: singular, bang-bang and mixed. All singular trajectories are shown to be optimal. We use reflection-symmetries of system (1) to restrict control in singular case as follows: \begin{align} |u_1 (t)| \leq 1, \qquad u_2 \equiv 1. \quad (4) \end{align} Geometric formulation of Pontryagin maximum principle is applied to system (1),(2) with condition (4) in order to describe boundary of the attainable set via singular trajectories (this set coincides with the part of the sub-Finsler sphere filled by singular trajectories). This set is explicitly described; we prove that it is semi-algebraic. The similar results are obtained also for a one-parameter family of left-invariant sub-Finsler problems on the Engel group. This study is part of joint work with Yu.~L.~Sachkov and E. Le Donne on sub-Finsler problems on Engel and Cartan groups.

By a branched cover we refer to a continuous, open and discrete mapping, and the set of points where it fails to be locally injective its branch set. By the classical Stoilow Theorem, a branched cover between planar domains is locally equivalent to the winding map and the equivalence is even quasiconformal when the original mapping is quasiregular. In higher dimensions the claim is not true, except for some special cases. Indeed, by the classical theorems of Church-Hemmingsen amd Martio-Rickman-Väisälä, a branched cover between euclidean n-domains is locally equivalent to a winding map when the image of the branch set is an n-2 -dimensional hyperplane.

Gromov introduced a good definition of limits of sequences of pointed metric spaces. Hence, we have a notion of tangent metric spaces. The objects of the talk are the metric spaces that have the same metric tangent at every point. First, we shall characterize among finite dimensional spaces what are such tangents: for geodesic spaces we only have subFinsler Carnot groups; in general, we still have graded nilpotent Lie groups. Second, we discuss some properties that one can deduce on the initial metric space from its tangents, possibly assuming that the tangents are `uniformly close’, in the sense that the convergence of the dilated metric spaces is uniform on the point that is chosen as base point. In this case, for example, we have that the doubling dimension is locally given by the dimension of its metric tangents. The main discussion will be whether there are good maps between the tangents and the original space. For example, every n-regular subset of R^N with uniformly close tangents isometric to Euclidean n-space is uniformly rectifiable. The situation for non-Abelian tangents is different. Indeed, on the one hand there exist subRiemannian Lie groups for which there are no quasiconformal maps between their open subsets and the tangents. On the other hand, we shall describe a recent result stating that if M is a sub-Riemannian manifold with tangent N at every point, then all of M except for a null set can be covered by countably many bilipschitz maps defined on subsets of N. We shall end with open problems on other settings: a) spaces with normed vector spaces as tangents b) submanifolds of Carnot groups that are intrinsically C^1. The presented results comes from a collection of works obtained in collaborations with M. Cowling, G.C. David, V. Kivioja, S. Li, S. Nicolussi Golo, A. Ottazzi, T. Rajala, B. Warhurst, and R. Young.

In the absense of differential forms, metric currents were defined by Ambrosio and Kirchheim to be certain multilinear functionals acting on Lipschitz functions. In this talk I define polylipschitz forms, which can be thought of as a metric substitute for differential forms, and illustrate their duality with metric currents. Time permitting, I will explain how polylipschitz forms can be used to construct a pull-back of metric currents by BLD-maps, a subclass of Lipschitz maps. The pull-back is, in a sense, a right inverse of the push-forward of currents. (The latter can be defined for any lipschitz map.) Joint work with P. Pankka

In the study of rough paths in stochastic PDE theory to each path in $\mathbb R^n$ one assigns its signature via the method of iterated integrals. With the purpose of metrizing the space of signatures as limits of sub-Riemannian groups we formalize the notion of limit of an inverse system of metric spaces with 1-Lipschitz projections having unbounded fibers. Hambly-Lyons's result on the uniqueness of signature implies that the space of signatures is a geodesic metric space homeomorphic to a tree that brunches at every point with infinite valence. As a particular consequence we deduce that every curve in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step. Joint work with Roger Zuest

Classically an isometric embedding between normed spaces must be affine if the norm on the target is strictly convex. The topic of this talk is the analogous result for Carnot groups of step 2.

TBA

The De Giorgi's rectifiability theorem states that the reduced boundary of a set of finite perimeter in a Euclidean space is rectifiable, i.e., it can be covered by a countable union of C^1 hypersurfaces, up to a set of codimension-one Hausdorff measure zero. The same results has been generalized by Franchi, Serapioni and Serra Cassano in all Carnot groups of step 2 and by Marchi to Carnot groups of type *. In higher generality, the validity of this result is still unknown and only partial result are available. We show that the reduced boundary of a set of finite perimeter in a general Carnot group has a "cone property" which implies a Lipschitz-type rectifiability in a class of Carnot groups which includes e.g. the Engel group and more in general all filiform groups of the first kind. This research has been done in collaboration with Enrico Le Donne, Terhi Moisala and Davide Vittone.

Rectifiability of sets of finite perimeter is one of the biggest open problems of Geometric Measure Theory in the setting of Carnot groups. The main issue for applying De Giorgi's proof from the Euclidean setting is that in Carnot groups, the blow-ups of sets of finite perimeter may have wilder structure than in the Euclidean case. I will give some new sufficient and necessary conditions for Carnot groups where the blow-up is always a half-space, and hence the De Giorgi's proof can be applied.

In this talk we plan to discuss compact metric spaces that admit a family of uniformly strongly quasi-Moebius self-homeomorphisms acting transitively on triples of distinct points. When all maps from this transitive family are Moebius, complete characterizations are available. However, the situation becomes more complicated when dealing with strongly quasi-Moebius maps.

Carpets are metric spaces that are homeomorphic to the standard Sierpinski carpet. By a theorem of Whyburn, they have a natural topological characterization. Consequently, they arise in many contexts involving dynamics, self similarity or geometric group theory. In these contexts, the Loewner property would have many structural and geometric implications for the space. However, unfortunately, we do not know if the Loewner property would be satisfied, or even could be satisfied, in many of the applications of interest. I will discuss explicit constructions infinitely many Loewner carpets with arbitrary conformal dimension, and whose snow-flake embeddings in the plane are explicit. I will further discuss rigidity results that ensure that even for the same conformal dimension we have infinitely many quasi symmetrically distinct carpets. The first part will present the general setting, construction and results. The latter part will give proofs of some of the parts and introduce the main technical tools used.