Minimal surfaces in the roto-translation group.

At the center of all of the applications to neuroscience and image processing is the study of the mathematics of minimal surfaces in the roto-translation group and, more broadly, in different sub-Riemannian spaces. 

 

Initial work done by myself and Nicola Garofalo showed that sufficiently smooth minimal surfaces in the Heisenberg group (a sub-Riemannian space closely related to the roto-translation group) can be seen as surfaces in three dimensional Euclidean space that satisfy a strong geometric condition:  they are ruled surfaces with rules picked from a special family of lines.  This work was extended by my former graduate student, Dan Cole, to the Martinet space and, more recently to the roto-translation group (by myself and Rob Hladky).

 

Undergraduate projects in mathematics would build on this characterization.  For example, projects might include:

 

1. Constructing examples of minimal surfaces in the roto-translation or Heisenberg groups with a variety of interesting geometric or topological properties.

2. Investigating gluing techniques to build examples of minimal surfaces that are not smooth.

3. Any investigation in this area would have direct application to either neuroscience or image processing (or both) and could be combined with one of those areas.

 

Experience and requirements:  Students interested in these (or other similar projects) should have completed our calculus sequence, linear algebra and at least one analysis course (preferably math 63).  In addition, students with knowledge of partial differential equations would probably find that a plus.