Associate Professor of Mathematics
Rose-Hulman Institute of Technology
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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 in-painting.