Seminar talk, 30 November 2022

From Geometry of Differential Equations
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Speaker: Dmitry Talalaev

Title: Zamolodchikov Tetrahedron equation

The main subject of the talk is the Zamolodchikov tetrahedron equation, which is the next n-simplex equation after the Yang-Baxter equation. This equation finds its embodiments in the theory of cluster manifolds, exactly-solvable models of statistical physics in dimension 3, as well as the theory of invariants of 2-knots, that is, classes of isotopies of embeddings of a two-dimensional surface in a 4-dimensional space.

The main focus of the report will be on the definition of this class of equations in terms of the hypercube face coloring problem, the cohomology complex associated with each solution of the n-simplex equation. We will discuss how these definitions are realized in the case of n=3, that is, in the case of the tetrahedron equation, and some interesting classes of solutions to this equation arising in modern mathematics.