Non-linear quotient mappings between Banach spaces.
A mapping $f$ from a Banach space $X$ to a Banach space
$Y$ is called a uniform quotient mapping if it is uniformly
continuous and for every $x\in X$ the image of
a ball with center $x$ and radius $r$ contains
a ball with center $f(x)$ and radius $\delta(r)>0$,
where $\delta$ is independent of $x$.
If the mapping is Lipschitz and $\delta(r)\ge cr$
for some $c>0$ independent of $r$ we call the mapping
a Lipschitz quotient.
These notions seems to be the right analogues of linear quotient
maps. They can also be viewed as quantitative versions of
continuous open mappings.
In this series of talks I'll survey some recent series of works
done in collaboration with Johnson, Preiss and Lindenstrauss
(and some also with Bates) revolving around the question: to what
extent does the existence of a uniform or Lipschitz quotient mapping
between $X$ and $Y$ ensure the existence of a linear quotient mapping.