Hamiltonian Systems

Discretization of the notion of a holomorphic function is one of the central problems in discrete geometry. In this talk, we will discuss two such discretizations. The first is given by discrete conformal maps, i.e. maps from the square lattice to the plane such that the image of every unit lattice square is a harmonic quadrilateral, while the second is given by Schramm’s orthogonal square-grid circle patterns.

We will discuss how these two models are related from the perspective of Poisson geometry, and how this relationship can be used to show that both models are, in an appropriate sense, completely integrable systems. Based on joint work with M. Arnold.

https://arxiv.org/abs/2607.08901