Frequency synchronization conditions for coupled oscillators on lattices
Tianqi Wu

In this talk I will report new synchronization conditions for Kuramoto
oscillators coupled through a d-dimensional lattice. Kuramoto model is one
of the most representative models in which we study the phenomenon of
collective synchronizations. I will briefly review pioneering and
important results on the phase cohesiveness in classical Kuramoto model,
and then shift to the new setting on the frequency synchronization
problems for Kuramoto oscillators on general finite graphs. We will report
the first frequency synchronization condition which is specially for the
lattices. Additionally, by viewing the lattice as the discretization of
the space, we achieved a novel continuous space model, as well as the
continuous synchronization condition analogue to the previous one. To
prove those two synchronization conditions, it amounts to show the
existence of solutions to some discrete (resp. continuous) elliptic PDE of
divergence form. The two main ingredients in the proof are variational
methods and gradient estimates for discrete (resp. continuous) elliptic

The first hour (4 - 5pm): we will give the introduction and present all
the results and indicate the steps in the proofs.
The second hour (5:05 - 6pm): we will fill more details about the proof.