Click the title above to play a fun educational computer game. Its goal is to visualize what is called Nielsen's algorithm in geometric group theory. In its classical definition, Nielsen's algorithm is stated in an unrevealing algebraic form. We created a visual, geometric way of describing and running Nielsen's algorithm: the player draws several paths in a tree and then transforms the paths by inverting them and concatenating.
Read the article explaining the mathematics behind this game: Back and forth the Nielsen theorem.
The game comes in four parts, corresponding to the rank of the free group: rank 1, rank 2 and rank 3, as well as the dual version of rank 2 game. In rank 1, as you will see, Nielsen's algorithm can be viewed as a generalization of the euclidean algorithm for finding greatest common divisors.
In Spring 2026 (as an IML project):
Professor: Igor Mineyev.
Graduate mentor: Hui Nguen.
Undergraduate students: Eunice Choi, Prabhat Kalle,
Amy Kodama,
Zainab Memon.
In Fall 2025 (as an IML project):
Professor: Igor Mineyev.
Graduate mentor: Jihong Cai.
Undergraduate students:
Eunice Choi, Prabhat Kalle, Nicola Lin, Maxwell Liu, Khushi Mamtani, Bowen Xiao.
In Summer 2025 (informally):
Professor: Igor Mineyev.
Graduate mentor: Jihong Cai.
Undergraduate students: Maxwell Liu.
In Spring 2025 (as an IML project):
Professor: Igor Mineyev.
Graduate mentor: Jihong Cai.
Undergraduate students: Haochen Ding, Maxwell Liu, Diya Mathew, Ruoqi Wang.
The link to this and other games is available from Igor Mineyev's Math Page.