1. Extensions

Linear dependence of constants in biLipschitz extension theorems
Problem 1.1.
[Leonid Kovalev] Do biLipschitz extension theorems admit linear dependence on constants? 
Dependence on parameters
Problem 1.2.
[Charles Smart] Do canonical Lipschitz extensions exist which depend nicely on parameters? 
Deterministic proof of the Lee–Naor Lipschitz extension theorem
The Lee–Naor theorem states that, given a doubling subset $A$ of a metric space $X$ and an $L$Lipschitz map from $A$ to a Banach space $Y$, there exists a $C(\log m)L$Lipschitz extension $X\to Y$. Here $m$ is the doubling constant of $A$.Problem 1.3.
[Piotr Hajlasz] Is there a deterministic proof of the Lee–Naor Lipschitz extension theorem giving the sharp dependence on the doubling constant?
Remark. There are deterministic proofs of similar results by Lang–Schlichenmaier and Brudnyi–Brudnyi.


Absolute Lipschitz retract constructions
${\mathbb Q}_Q(Y)$ denotes the space of unordered $Q$tuples $y=[y_1,\ldots,y_Q]$ of elements of $Y$ equipped with the metric $d(y,z) = \min \{ \max \{ d(y_j,z_{\sigma(j)}) \} : \sigma \in S_Q \}$, where $S_Q$ denotes the symmetric group on $Q$ letters.Problem 1.4.
[Thierry De Pauw] If $Y$ is an absolute Lipschitz retract, is the same true of the space ${\mathbb Q}_Q(Y)$?
Remark. [Leonid Kovalev] The space ${\mathbb Q}_Q(Y)$ is similar to the $Q$th symmetric product space; similar questions could be of interest for symmetric products.


Banach space pairs with the Lipschitz extension property
Problem 1.5.
[Urs Lang] Does the pair $(L^2,L^1)$ have the Lipschitz extension property? 
Absolute Lipschitz retracts
Problem 1.6.
[Urs Lang] Are all Hadamard manifolds absolute Lipschitz retracts? 
Lipschitz homotopy groups of the Heisenberg group
${\mathbb H}^n$ denotes the $n$th Heisenberg group, equipped with the CarnotCarathéodory metric. $\pi_k^{\scriptstyle{Lip}}(X)$ denotes the $k$th Lipschitz homotopy group of a metric space $X$. It is known that $\pi_n^{\scriptstyle{Lip}}({\mathbb H}^n) \ne 0$ (DeJarnette–Hajlasz–Lukyanenko–Tyson) [arXiv:1109.4641].
Remark. Solved by Hajlasz, Tyson, Schikorra, Wenger, Young... in HOMOTOPY GROUPS OF SPHERES AND LIPSCHITZ HOMOTOPY GROUPS OF HEISENBERG GROUPS as well as "Lipschitz Homotopy Groups of the Heisenberg Groups", to appear in GAFA (Geometric and Functional Analysis) in February 2014.


Lipschitz homotopy groups of the Heisenberg group II
Problem 1.8.
[Piotr Hajlasz and Jeremy Tyson] Does $\pi_n^{\scriptstyle{Lip}}({\mathbb H}^n)$ contain torsion elements?
Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.