Abstract: Integrability in statistical physics models usually means that the partition function can be represented in terms of the transfer matrix included in the "large" commutative family. The last property for two-dimensional models is traditionally accompanied by the structure of a vertex model with a weight matrix satisfying the Yang-Baxter equation. This talk is about the generalisation of this idea to a larger dimension, in particular, I will consider the three-dimensional Ising model, as well as the Hopfield neural network model on a 2-dimensional triangular lattice in the memory phase. It turns out that both these models have a vertex representation, with a matrix of weights that satisfies the deformation of the generalization of the Yang-Baxter equation in 3 dimensions, the so-called twisted equation of tetrahedra. In both cases, the combinatorics of the hypercube is essentially used to construct the matrix of weights.

Location: 3 Komsomol'skaya st., B. Delaunay "Discrete and computational geometry" laboratory.