Talk by Daniel Parra
On May 4th, 2023, Daniel Parra (University of Santiago, Chile) gave a talk about "Continuum limit for a discrete Hodge-Dirac operator on square lattices" as part of the online workshop on "Dirac equation between discrete and continuous: new trends and applications" organized by Ginestra Bianconi (Queen Mary University of London) and Delio Mugnolo (FernUniversität in Hagen).
We study the continuum limit for Dirac-Hodge operators defined on the n dimensional square lattice hZ^n as h goes to 0. This result extends to a first order discrete differential operator the known convergence of discrete Schrödinger operators to their continuous counterpart. To be able to define such a discrete analog, we start by defining an alternative framework for a higher–dimensional discrete differential calculus to the standard one defined on simplicial complexes. We then express our operator as a differential operator acting on discrete forms to finally be able to show the limit to the continuous Dirac-Hodge operator.