Spring 2022

Ping Pong Patches

Students: Jordan Grant, Jeremy Krill, Samuel Perales
Mentor: Teddy Weisman
Faculty advisor: Jeff Danciger

This project finalized the work of the two previous semesters to create an algorithm which can computational generate proofs of a representations faithfulness in \(\mathrm{SL}(2, \mathbb{R})\). The work includes a demo in the form of a web application of proofs for a cyclic free product, triangle group, and surface group.

Fall 2021

Stable commutator length

Students: Simon Xiang, Jimmy Xin, Ruiqi Zou
Faculty advisor: Lvzhou (Joe) Chen

Billiards in the projective plane

Students: Andrew Bacon, Henry Castillo, Vincent Solon
Mentor: Charlie Reid
Faculty advisor: Jeff Danciger

Automatic ping-pong

Students: Jordan Grant, Jeremy Krill, Samuel Perales
Mentor: Teddy Weisman
Faculty advisor: Jeff Danciger

This project studied representations of strongly geodesically automatic groups in \(\mathrm{SL}(2, \mathbb{R})\) using a generalized form of the ping-pong lemma. It extended work from the previous ping-pong project to develop an algorithm that searches for an upper-bound on the kernel of a representation.

Spring 2021

Convex domains in projective geometry

Students: Sarah Bruce, Maxwell Nakos, John Teague, Anabel T. To
Mentors: Florian Stecker, Neža Žager Korenjak
Faculty advisor: Jeff Danciger

This project explored fundamentals of projective geometry and properties of convex subsets of \(\mathbb{R}P^2\). It produced some interactive demos illustrating the Benzecri cocompactness theorem, an important result in convex projective geometry. View them here!

Ping-pong and beyond

Students: Jordan Grant, Abhay Katyal, Samuel Perales
Mentors: Max Riestenberg, Teddy Weisman
Faculty advisor: Jeff Danciger

This project studied free groups in \(\mathrm{SL}(2, \mathbb{R})\) using the ping-pong lemma. It involved using the lemma to design and implement an algorithm that decides if a given set of matrices generate a free group. See a demonstration here!