Undergraduate Research

My mathematical research has recently focused on the so-called minimal surface problem in sub-Riemannian geometry.  Roughly, sub-Riemannian geometries are those that model spaces with limited degrees of freedom.  The minimal surface problem is the analogue of the classical minimal surface problem in three-dimensional space.

 

Recently, this rather esoteric corner of mathematics has met with a new and interesting application — a model of the function of the first layer of the visual cortex (V1) in the brain (due to G. Citti and A. Sarti at the University of Bologna) naturally yields a sub-Riemannian geometry called the roto-translation group.  Moreover, using this model, Citti and Sarti showed that V1’s role in solving the occlusion problem, i.e. the way that the brain completes partially hidden objects, involves the solution to the minimal surface problem in the roto-translation group.  These solutions are the model’s prediction of how the first layer of the cortex contributes to the completion of occluded objects.

 

The occlusion problem has natural ties to digital image processing.  For example, a digital image may be obscured or corrupted by some flaw leading to an occlusion and require repair.  Or, in training a computer to recognize objects within a scene, the completion of image elements partially occluded by other elements  leads to better pattern recognition. 

 

The opportunities for undergraduate research lie in these three overlapping problems.  To learn more about each aspect click on the links below:

 

Minimal surfaces in the roto-translation group.

Interactions with neuroscience.

Digital occlusion problems.

 

Within each of these topics are projects suitable for a wide range of interests and goals, including senior theses, independent studies/reading courses, off-term or on-term research projects, E. E. Just Fellows, Presidential scholars, Senior Fellowships, etc. 

 

Initially, research projects will be available for those students with substantial mathematical experience.  In the 07-08 academic year, I intend to develop projects suitable for the WISP program and for interested first and second year students. 

 

Using mathematics to model problems in visual occlusion