Geometric PDE and Applied Analysis Seminar (April 15, 2026)

Title: Horizontal mean curvature flow in the Heisenberg group as scaling limit of an interacting particle system

Speaker: Prof. Nicolas Dirr (Cardiff University)

Abstract: We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a microscopic (i.e. cellular) level to macroscopic models of image processing through a multi-scale approach. The nonlocal equation, which is very similar to the Ermentraut-Cowan mean field population model, can be derived from an interacting particle model and its discretized version. As sub-Riemannian geometries play an important role in the Citti-Sarti-Petitot model of the visual cortex, this paper provides a mathematical framework for a rigorous upscaling of models for the visual cortex from the cell level via a mean field stage to curvature flows which are used in image processing. We present some numerical results comparing the model to a known exact solution. For different choices of the parameters, the numerical algorithm can be connected to a Bence–Merriman–Osher scheme for surface evolution or a convolutional neural network. Joint work with Giovanna Citti, Federica Dragoni and Raffaele Grande.