Abstract：

The tetrahedron equation, a three-dimensional analogue of the Yang-Baxter equation, is the fundamental relation in studying three-dimensional integrable systems. In this talk, I will discuss the construction of solutions of the tetrahedron equation using cluster algebra, based on work with my supervisor, Junya Yagi (arXiv:2211.10702) and further developed with Rei Inoue, Atsuo Kuniba, Yuji Terashima, and Junya Yagi (arXiv:2403.08814). Our cluster algebraic approach reproduces most known solutions of the tetrahedron equation. I will also briefly talk about how these solutions can be identified with the partition functions of certain 3d N=2 gauge theories on a squashed sphere.