In this GeoMeG meeting, we discuss differentiability of Lipschitz functions that are defined on a separable Banach space. We follow a paper of Aronszajn, 1976, where he proves that Lipschitz maps are differentiable up to a certain exceptional class. We see that a suitable notion of differentiability is just slightly weaker than the Fréchet differentiability and that in the finite-dimensional case, the exceptional sets coincide with the sets of Lebesgue-measure zero.

In this presentation we discuss recent develoments and open problems in the theory of robust output tracking and disturbance rejection for linear infinite-dimensional systems. The presented theory can in particular be used in designing robust error feedback controllers for linear partial differential equations. The controllers solving the robust output tracking problem are characterized by the fundamental "Internal Model Principle" first presented for finite-dimensional systems in the 1970's. As our main results we present a generalization of the Internal Model Principle for infinite-dimensional linear systems. Finally, we discuss different internal model based controller structures for selected classes of linear partial differential equations.

In this GeoMeG meeting, following the paper 'On homogeneous manifolds of negative curvature' by Ernst Heintze I will show that such manifolds are exactly certain solvable Lie groups, nowadays called Heintze groups. These Lie groups are constructed from nilpotent Lie groups admitting a dilation as a semidirect product with reals.

Which metric spaces can be parametrized by some Euclidean space under a quasiconformal or quasisymmetric mapping? We discuss some ongoing work in this area of research.

In this GeoMeG meeting, I will survey the question of classifying (connected) simply connected nilpotent Lie groups up to quasi-isometry and present the known QI-invariants provided by Pansu, Shalom, and Sauer. Following Cornulier, these invariants are to be compared with the classification of nilpotent Lie algebras (hence of simply connected nilpotent Lie groups) up to dimension 7. The role of Lie algebra cohomology and the related spectral sequence will be discussed as they are the source of potential QI-invariants. If time permits or in a future meeting, we shall compare this to recent QI-rigidity results of Amann.

I will discuss the self-improvement result of Keith-Zhong and a new proof. The first half of the talk will give an overview of the argument and the main steps, as well as the crucial tools that are used. The latter half will go more into detail of how the tools are used, and I will mention a couple other applications of the same ideas.

The volume entropy of a metric measure space is the exponential growth rate of volumes of balls. A recent result of N. Tholozan shows that, in the context of Hilbert geometries, entropy can never exceed the hyperbolic entropy (n-1 in dimension n). This result is absolutely not rigid and in fact, maximal entropy is achieved as soon as the boundary is sufficiently regular. This leads to the general question on the relation between the regularity of the convex set and the value of the volume entropy. The aim of this talk is to present two results which state that the relation does exist. This is a joint work with J. Cristina. In a first part of the talk, I'll carefully explain what are Hilbert geometries and try to give a feeling on the behavior of volume entropy in those spaces.

In this talk I will discuss a method that permits any metric space $(X,d)$ to be continuously injected into a compact topological space $\overline{X}_{d}^{h}$, the so-called metric compactification. This generalizes the notion of Gromov's horofunction compactification which is mainly studied on proper geodesic metric spaces. I will give a complete description of the metric compactification of the classical Banach spaces $\ell_{p}$ in finite and infinite dimensions. Also I will describe the metric compactification of the Hilbert geometry on the $n$-simplex.

(joint work with Assaf Naor) The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is hard to embed in $\mathbb{R}^n$. Cheeger and Kleiner introduced a new notion of differentiation that they used to show that it does not embed nicely into $L_1$. This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques based on uniform rectifiability that let us find sharp bounds on the distortion of embeddings of $\mathbb{H}$ and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem. $$ \ $$ In the first part of the talk, we will give some background on the connection to the Sparsest Cut Problem, an overview of previous work, and a sketch of our results. In the second part, we will go into more detail about uniform rectifiability and corona decompositions in the Heisenberg group and how we use them to prove our results.

We discuss a general Sard-type rigidity phenomenon for Lipschitz maps in metric measure spaces. We will then discuss new results of the speaker and Kyle Kinneberg concerning mappings between metric measure spaces and Carnot groups, addressing questions of Semmes. If time permits, we will also discuss an application to hyperbolic geometry.