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Research Opportunities for Undergraduates (Past Projects)

### Hyperbolic geometry and Riemann surfaces, and Systolic geometry

Mentor: Bjoern Muetzel

If you are an undergraduate interested in a reading course, independent study or working on a research project, feel free to contact me. I am particularly interested in the following topics.

The hyperbolic plane is a space of constant negative curvature minus one, where different rules than in Euclidean space apply for geodesics, the geometry of polygons and the area of disks. A hyperbolic surface can be seen as a polygon in the hyperbolic plane with identified sides. We call such a surface a Riemann surface. Many questions about Riemann surfaces are still open or under study. Hyperbolic geometry is used in the theory of special relativity, particularly Minkowski spacetime.

A systole of a surface is a shortest non-contractible loop on a surface. Every surface has a genus $$g$$, where informally $$g$$ denotes the number of holes. Surprisingly given any surface of fixed genus $$g$$ and area one, the systole can not take a value larger than $$c \cdot \frac{\log(g)}{ \sqrt{g}}$$, where $$c$$ is a constant. A large number of families of short curves on surfaces satisfy this upper bound and example surfaces can be found among the hyperbolic Riemann surfaces.

### Numbers of nodal domains in quantum chaotic billiards

Undergrad research project with Prof Alex Barnett. October 2010.

I am looking for someone to work with me on a numerical research project in quantum chaos, probably starting in the new year, or next summer, described below. It's a question that some well-known people in the field of quantum chaos and number theory consider important, but that will require some dexterity in figuring out how to make use of some codes I have written, and gather the data (ie, experience with C or Matlab, etc). Anyway, it could make a nice senior thesis, or a shorter project, that has some good `bang for the buck'. Get in touch with me if interested.

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.

### Analysis of Protein Structure

Presidential Scholar Rob Taintor '08 worked with professors Rockmore and Leibon, implementing a fast Hermite Transform algorithm that will be used in analyzing data to help determine protein structure. This work is currently being written up for journal publication.

### Topological invariants of wave fronts and virtual knots

Presidential Scholar Evarist Byberi '08 worked with professor Chernov on the project titled "Topological invariants of wave fronts and virtual knots." The result of this project were new interesting results about bridge and unknotting numbers of virtual knots.

### Medical Imaging

Chetan Mehta '08 worked on optimal measurements for Diffuse Optical Tomography, a medical imaging inverse problem. Senior Thesis, advisor --- prof. Alexander Barnett.

### Chaotic Dynamics

Presidential Scholar Chor Lam '09 worked with professor Barnett on chaotic dynamics in the mushroom billiard (Fall-Winter 2007-2008).

### Acoustics/Number Theory

WISP student Vissuta Jiwariyavej '09 analysed the acoustic impulse-response of the racquetball court using method of images and techniques from number theory (prof. Alexander Barnett).

### A Mathematical Analysis of Concussive Injuries and their Diagnostic Tools

Michael P. McClincy '06 studied head injuries in sports and the tests used to assess their mental effects, and even devised a combined score which was a better predictor than the individual scores. Senior Thesis, advisor --- prof. Peter Winkler.