Singular Yamabe Problem:
Given a compact Riemannian manifold (M,g)
of dimension n and a nonempty closed subset G,
find a complete metric h with constant scalar curvature
on M\G that is pointwise conformally
related to g. 
Shape of a Cookie:
Given
the contact curve of a cookie and the cookie sheet, determine
the shape of the cookie. (Possibly a nonsymmetric sesile drop question
in terms of capillary phenomena.) 
Bicycle Tracks:
Some
questions involving the geometry of bicycle tracks
Given a pair of tracks on a surface, is it
possible that they were produced by a bicycle?
Can a bicycle
create a single tire track?,
Can you create
a pair of tire tracks for which it is impossible to tell which
way the bicycle is going?,
Can you create a pair of tire tracks that can
be constructed by bicycles of different sizes? 
PDEs in Image Processing:
Application of PDEs to Image processing
problems, specifically denoising, filtering, and inpainting. 
