Geometry Lab

Computational problems in hyperbolic geometry

Since Spring 2025, I have been leading a group of 3 undergraduates (Joseph Hummel, Kaius Noble, and Prajith Velicheti) on some experimental/computational research work through the Texas Experimental Geometry Lab. Our project began by working on producing visualizations for Markov codings for geodesic flows on hyperbolic surfaces, resulting in some very nice animations and exposition available on the original project website.

Once that portion of the project was completed, we moved on to working on a computational optimization problem in hyperbolic geometry, initially proposed by my advisor (Jeff Danciger). Our efforts there resulted in an algorithm to locate and certify in finite time which curve is maximally stretched between two hyperbolic structures on a surface. The results of our initial research project were presented by the undergraduates at the Geometry Labs United IV conference, hosted at UW Madison in the Summer of 2026. We describe the methods and approach in a poster (also embedded below) as well as presentation slides, both of which were presented at the aforementioned conference.

We are finishing up a research paper titled Certifying Maximal Stretch Ratios for Fuchsian Free Group Representations, to be submitted for publication shortly. We are also continuing to work on extending the scope of our methods to more delicate situations. Our work may yet result in another article, as we are close to resolving some more questions of research interest.

My role in the project has been as a manager and mentor for the undergraduates. While I did not write by my hand any of the code involved, I worked closely with our undergraduates on the design of the algorithms and implementation choices. Furthermore, I helped to drive the theoretical research component alongside my advisor, writing proofs to demonstrate the sufficiency of the computations carried out in code. I mentored the students through teaching them necessary background for the project, then worked with them on formulating and probing the research question via computations in code, and finally on writing up the results in a paper.