#
Mapping theory in metric spaces

January 9 to January 13, 2012
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Luca Capogna,
Jeremy Tyson,
and Stefan Wenger

## Original Announcement

This workshop will be devoted to mappings between metric spaces. Recent years have witnessed remarkable advances on basic questions concerning uniqueness, extendability, embeddability, uniformization and extremality of mappings in a variety of regularity classes. A persistent source of complexity is the influence of the geometry of the target space: problems whose solution is straightforward in the scalar-valued case become more intricate for vector-valued targets, even more so in case the target is a (nonlinear) manifold or metric space.
The purpose of this workshop is to bring together researchers with differing backgrounds and expertise to highlight common techniques and methods and leverage existing knowledge towards the successful resolution of interdisciplinary problems.
The main topics for the workshop are:

- Sobolev mappings on and between metric spaces
- Lipschitz and optimal Lipschitz extensions
- bi-Lipschitz and quasisymmetric embedding and uniformization
- mass transportation

The workshop will focus on particular problems of contemporary importance such as: (i) the role of quasiconformal geometry in the uniqueness problem for vector-valued absolute minimizing Lipschitz extensions, (ii) the interplay between bi-Lipschitz embedding theory and the Lipschitz density problem for metric space-valued Sobolev mappings, (iii) rigidity and uniformization problems for bi-Lipschitz or quasisymmetric mappings between metric spaces, (iv) extremal problems for quasiconformal mappings and their connection to hyperelasticity.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop: