We discuss recent joint work with Guy C David. Any Lipschitz map from R^n can locally be approximated by a linear map. A 1988 theorem of Jones says that the domain can be broken up into a finite number of pieces, where the map is biLipschitz, except for a piece with small image content (in dimension n). If the dimension of the image is smaller, then a 2012 theorem (joint with J. Azzam) shows that there is a large piece on which, after a global biLipschitz change of coordinates, the map behaves like a projection. With Guy C David, we recently improved upon this, getting that the domain may be broken up into pieces exhibiting this nice behavior, except for a piece with a `small’ image (in the appropriate sense). All results above are quantitative! We will discuss the above results, a construction by Kaufman showing it is sharp, and a very recent result showing this Kaufman construction is the only way to get such an example.