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Research Opportunities for Undergradates



Numbers of nodal domains in quantum chaotic billiards
(Optimally Winter/Spring/Summmer 2011)

The highly-excited vibrational modes of a drum (or quantum billiard) have regions of positive and negative motion that divide the surface into so-called nodal domains. In the last few years Bogomolny-Schmit proposed a percolation model which predicts the number and variance of these domains, but very few tests of this have been done using actual systems. The project would be to do a large-scale, and possibly publishable, numerical study of the numbers of nodal domains in chaotic billiards, and Maass forms. Both are of current interest to mathematicians---in particular number theorists such as Peter Sarnak. Codes exist for the modes; you will need to interface to them for the data collection, so programming experience (eg, C or Matlab) is essential.

Advisor: Dr Alex Barnett Rm 206, Kemeny Hall

Analysis of the Currency Markets

In this project the currency exchange market will be analyzed from a network theoretic point of view. Necessary skills include some basic statistics and ability to mine relevant data from the web using the appropriate software.

Advisors: Professors Rockmore, Pauls, and Leibon.

Minimal surfaces in the rototranslation group

Recently, there has been a great deal of interest in the study of minimal surfaces in a new setting, that of sub-Riemannian geometry. Projects in this area would include extending and developing techniques for the construction of minimal surfaces in the roto-translation group.

More information/requirements

Advisor: Prof. Pauls

Applications of minimal surface theory to the neuroscience of vision

Interest in minimal surfaces in the rototranslation group stem from, in a large part, a new model of the function of the primary visual cortex. Building on work of Patrick Karas '07, we wish to develop perceptual tests and predictions from this model for the purposes of validation and revision.

More information/requirements

Advisor: Prof. Pauls

Minimal surface theory and digital image processing

A new model of the function of primary visual cortex via minimal surface theory provides a new approach to problems in digital image reconstruction. In this project, students would develop and implement (in MATLAB) image disocclusion algorithms based on this model.

More information/requirements

Advisor: Prof. Pauls

[Past projects]