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.

22/January/2018

Robust control of linear partial differential equations

(MaD 380)

Lassi Paunonen (Tampere University of Technology)

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.

29/January/2018

What is a Heintze group?

(MaD 380)

Ville Kivioja (University of Jyväskylä)

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.

5/February/2018

Quasiconformal and quasisymmetric uniformization of metric spaces

(MaD 380)

Matthew Romney (University of Jyväskylä)

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.

12/February/2018

On the classification of nilpotent Lie groups up to quasi-isometry and Lie algebra cohomology

(MaD 380)

Thibaut Dumont (University of Jyväskylä)

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.

12/March/2018

New proof for Keith-Zhong self-improvement for Poincare inequalities

(MaD 380)

Sylvester Eriksson-Bique (University of California Los Angeles)

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.

19/March/2018

On the Entropy of Hilbert Geometries of Low regularities

(MaD 380)

Louis Merlin (Universite du Luxembourg)

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.

21/May/2018

On the metric compactification of Banach spaces

(MaD 380)

Armando W. Gutiérrez (Aalto University)

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.

28/May/2018

Embeddings of the Heisenberg group, uniform rectifiability, and the Sparsest Cut problem

(MaD 380)

Robert Young (Courant Institute, New York University)

(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.
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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.

11/June/2018

Lipschitz and bi-Lipschitz maps from PI spaces to Carnot groups

(MaD 380)

Guy C. David (Ball State University)

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.

10/September/2018

Mapping of Finite Distortion: Compactness of the Branch Set

(MaD 380)

Rami Luisto (University of Jyväskylä)

Quasiregular mappings and mappings of finite distortion are natural generalizations of holomorphic mappings to higher dimensions. Whereas the pointwise derivatives of holomorphic mappings map circles to circles, QR-maps and MFD are defined by requiring that the differential maps balls to ellipsoids with controlled eccentricity.
Under certain mild integrability conditions, mappings of finite distortion are continuous, open and discrete, as are all quasiregular mappings by the Reshetnyak theorem. For continuous, open and discrete mappings between Euclidean n-domains the branch set, i.e. the set of
points where the mapping fails to be a local homeomorphism, has topological dimension of at most n-2 by the Cernavskii-VÃ¤isÃ¤lÃ¤ theorem.
For quasiregular mappings more properties for the branch set are known, but several important questions remain open.
In this talk we show that an entire mappings of finite distortion cannot have a compact branch set when its distortion is locally finite and
satisfies a certain asymptotic growth condition; $K(x) < o(\log(|x|))$. In particular this implies that the branch set of entire quasiregular
mappings is either non-compact or empty. We furthermore show that the growth bound is asymptotically strict by constructing a continuous, open and discrete mapping of finite distortion from the Euclidean $n$-space to itself which is piecewise smooth, has a branch set homeomorphic to the $(n-2)$ torus and distortion arbitrarily close to the asymptotic bound $\log(|x|)$.
The talk is based on joint work with Aapo Kauranen and Ville Tengvall.

18/September/2018

Quasi-Mobius Homogeneous Metric Spaces

(MaD 355 Tuesday)

David Freeman (University of Cincinnati Blue Ash College)

Please note the time and days have changed.
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In this talk we will investigate metric characterizations of the boundaries at infinity of rank-one symmetric spaces of non-compact type in terms of their homogeneity with respect to M\"obius self-mappings. In particular, we will present a new characterization of such boundaries up to isometric and/or snowflake equivalence. Time permitting, we will also discuss progress toward a characterization up to bi-Lipschitz and/or snowflake equivalence.

24/September/2018

Quasiconformal mappings on the Heisenberg group

(MaD 380)

Matthew Romney (University of Jyväskylä)

This is an expository talk on the topic of quasiconformal mappings on the Heisenberg group. We will first review the definition of the Heisenberg group and discuss how it arises as the boundary of complex hyperbolic space. Next, we discuss how a quasi-isometry of complex hyperbolic space induces a quasiconformal homeomorphism of the boundary. The talk concludes with an overview of the flow method of KorÃ¡nyi-Reimann for constructing non-trivial examples.

1/October/2018

Sub-Riemannian structures on Lie groups

(MaD 380)

Rory Biggs (University of Pretoria)

A sub-Riemannian structure on a smooth manifold $\mathsf{M}$ consists of a smooth non-integrable distribution $\mathcal{D}$ on $\mathsf{M}$ and a Riemannian metric $\mathbf{g}$ on $\mathcal{D}$. When the manifold $\mathsf{M}$ is a (real, connected, finite-dimensional) Lie group and the distribution $\mathcal{D}$ and metric $\mathbf{g}$ are invariant under left-translations, then we have a left-invariant structure. Such structures are basic models for sub-Riemannian manifolds and as such serve to elucidate general features of sub-Riemannian geometry as well as being a rich source of examples and counterexamples for questions and conjectures.
In this talk I give an introduction to the (left-invariant) sub-Riemannian structures on Lie groups. The sub-Riemannian structures on the Heisenberg groups are treated as a case study: I will discuss, in this context, the problem of classification, determining the isometry group, and the geodesic (or length minimizing) problem. This then provides a platform to also discuss some developing problems and topics (e.g., work being done on low-dimensional Lie groups as well as on certain, often nilpotent, classes of Lie groups). Time permitting, three further topics can be touched on in an extended session: (1) isometries and automorphisms of sub-Riemannian structures on low-dimensional Lie groups (2) immersions and totally geodesic subgroups, (3) submersions and extensions of left-invariant sub-Riemannian structures.