Given a metric space homeomorphic to a Euclidean space (or a

sphere, for example) it is natural to ask for geometric parametrization

with a homeomorphisms which respect the metric, for example, bilipschitz

or quasisymmetric maps.

In 1996 Semmes constructed metric spaces which are homeomorphic to the

standard Euclidean 3-space but not admit geometric parametrizations.

The construction was surprising, since the metric space is close to a

Euclidean space from the point of view of measure and geometry. In 2004

Bonk and Kleiner showed that metric two dimensional spheres satisfying

the same conditions are in fact quasisymmetric to the standard sphere.

Although there are no good geometric homeomorphisms from Semmes' space

to the Euclidean space, there exist other natural geometric mappings to

the Euclidean space by results of Heinonen and Rickman in 2004. These

mappings give Semmes' space a natural ``branched parametrization''.

Semmes' construction is based on a classical geometric topology. In this

talk I will discuss the joint work with Jang-Mei Wu on general framework

of geometric decomposition spaces and their quasisymmetric

(non-)parametrizability. I will also discuss recent work with Kai Rajala

and Jang-Mei on the existence of branched parametrizations in this

general setting.

When? | 01.04.2014 17:15 |
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Where? | PER 08 Phys 2.52 Chemin du Musée 3 1700 Fribourg |

Contact | Department of Mathematics isabella.schmutz@unifr.ch |