Increasing trees and Kontsevich cycles.
Instability of the eikonal equation and shape from shading
In the shape from shading problem of computer vision one attempts to recover the three-dimensional shape of an object or landscape from the shading on a single image. Under the assumptions that the surface is dusty, distant, and illuminated only from above, the problem reduces to that of solving the eikonal equation |Du|=f on a domain in . Despite various existence and uniqueness theorems for smooth solutions, we show that this problem is unstable, which is catastrophic for general numerical algorithms. ...
Involutions in mapping class groups of non-orientable surfaces.
We prove that the mapping class group and the pure mapping class group of closed non-orientable surfaces with punctures are generated by involutions.