 Grant Agreement  University of Jyvskäylä Press Release 
Enrico Le Donne
 Personal website 
Sebastiano Don
Laurent Dufloux
Thibaut Dumont
Rami Luisto
Matthew Romney
Francesca Tripaldi
Fares Essebei
Eero Hakavuori
Ville Kivioja
Terhi Moisala
The Jyväskylä Geometry seminar is on Mondays afternoons
in the Department of Mathematics and Statistics.
The meeting is usually from 14:15 to 16:00 in room MaD 380.
Please, check in advance the calendar below.
25/March/2019  Metric currents and polylipschitz forms 
(MaD 380)  Elefterios Soultanis (University of Fribourg) 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 pullback of metric currents by BLDmaps, a subclass of Lipschitz maps. The pullback is, in a sense, a right inverse of the pushforward of currents. (The latter can be defined for any lipschitz map.) Joint work with P. Pankka 
3/June/2019  TBA 
(MaD 380)  David Fisher (Indiana University Bloomington) 
10/June/2019  TBA 
(MaD 380)  David Freeman (University of Cincinnati Blue Ash College) 
See all current Seminars  Analysis Seminar 
Past Seminars: 2018  2017  (older) 
11^{th}16^{th} August 2019  Jyväskylä (Finland)
Jyväskylä Summer School  Courses in Geometry
 Website 
23^{rd} Sep  3^{rd} Oct 2019  Oxford (UK)
Pansu's Fest, for the 60th birthday of Pierre Pansu
 Website 
Past Events  
February/2018  SubRiemannian Geometry and Beyond 
February/2019  SubRiemannian Geometry and Beyond, II 
We warmly thank the CVGMT Team for the support and
for hosting the following papers at CVGMT server.
Metric Lie groups admitting dilations (2019)  
E. Le Donne, S. Nicolussi Golo We consider leftinvariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative oneparameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$, $\lambda\mapsto\delta_\lambda$, so that $ d(\delta_\lambda x,\delta_\lambda y) = \lambda d(x,y)$, for all $x,y\in G$ and all $\lambda>0$. First, we show that all such distances are admissible, that is, they induce the manifold topology. Second, we characterize multiplicative oneparameter groups of Lie automorphisms that are dilations for some leftinvariant distance in terms of algebraic properties of their infinitesimal generator. Third, we show that an admissible leftinvariant distance on a Lie group with at least one nontrivial dilating automorphism is biLipschitz equivalent to one that admits a oneparameter group of dilating automorphisms. Moreover, the infinitesimal generator can be chosen to have spectrum in $[1,\infty)$. Fourth, we characterize the automorphisms of a Lie group that are a dilating automorphisms for some admissible distance. Finally, we characterize metric Lie groups admitting a oneparameter group of dilating automorphisms as the only locally compact, isometrically homogeneous metric spaces with metric dilations of all factors. 

Blowups and blowdowns of geodesics in Carnot groups (2018)  
E. Hakavuori, E. Le Donne We study infinitesimal and asymptotic properties of geodesics (i.e., isometric images of intervals) in Carnot groups equipped with arbitrary subFinsler metrics. We show that tangents of geodesics are geodesics in some groups of lower nilpotency step. Namely, every blowup curve of every geodesic in every Carnot group is still a geodesic in the group modulo its last layer. With the same approach, we also show that blowdown curves of geodesics in subRiemannian Carnot groups are contained in subgroups of lower rank. This latter result can be extended to rough geodesics. 

Restricting open surjections (2018)  
J. Jaramillo, E. Le Donne, T. Rajala (Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas) We show that any continuous open surjection from a complete metric space to another metric space can be restricted to a surjection for which the domain has the same density character as the target. This improves a recent result of Aron, Jaramillo and Le Donne. 

A note on topological dimension, Hausdorff measure, and rectifiability (2018)  
G. David, E. Le Donne The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$dimensional Hausdorff measure of $X$, $H^n(X)$, is finite. Suppose further that the lower $n$density of the measure $H^n(X)$ is positive, $H^n(X)$almost everywhere in $X$. Then $X$ contains an $n$rectifiable subset of positive $H^n(X)$measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results of CsörnyeiJones. 

Toward a quasiMöbius characterization of Invertible Homogeneous Metric Spaces (2018)  
D. Freeman, E. Le Donne We study locally compact metric spaces that enjoy various forms of homogeneity with respect to Möbius selfhomeomorphisms. We investigate connections between such homogeneity and the combination of isometric homogeneity with invertibility. In particular, we provide a new characterization of snowflakes of boundaries of rankone symmetric spaces of noncompact type among locally compact and connected metric spaces. Furthermore, we investigate the metric implications of homogeneity with respect to uniformly strongly quasiMöbius selfhomeomorphisms, connecting such homogeneity with the combination of uniform biLipschitz homogeneity and quasiinvertibility. In this context we characterize spaces containing a cut point and provide several metric properties of spaces containing no cut points. These results are motivated by a desire to characterize the snowflakes of boundaries of rankone symmetric spaces up to biLipschitz equivalence. 

On the quasiisometric and biLipschitz classification of 3D Riemannian Lie groups (2018)  
K. Fässler, E. Le Donne This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with leftinvariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasiisometries and we compare the quasiisometric classification with the biLipschitz classification. On the other hand, we study the problem whether two quasiisometrically equivalent Lie groups may be made isometric if equipped with suitable leftinvariant Riemannian metrics. We show that this is the case for threedimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric' is not a transitive relation. 

Please, contact the PI if you are interested to do a Ph.D. or a postdoc
in the framework of GeoMeG ERC project.
New positions will be open in the next years.
Prof. Enrico Le Donne
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