We consider a nonlinear flow on simplicial complexes related to the simplicial Laplacian, and show that it is a generalization of various consensus and synchronization models commonly studied on networks. In particular, our model allows us to formulate flows on simplices of any dimension, so that it includes edge flows, triangle flows, etc. We show that the system can be represented as the gradient flow of an energy functional, and use this to deduce the stability of various steady states of the model. Finally, we demonstrate that our model contains higher-dimensional analogues of structures seen in related network models.

arXiv link: https://arxiv.org/abs/2010.07421

 We generalize the recent holographic correspondence between an ensemble average of free bosons in two dimensions, and a Chern-Simons-like theory of gravity in three dimensions, by Afkhami-Jeddi et al and Maloney and Witten. We find that the correspondence also works for toroidal orbifolds, but we run into difficulties generalizing to K3 and Calabi-Yau sigma models. For the case of toroidal orbifolds, we extend the holographic correspondence to averages of correlation functions of twist operators by using properties of rational tangles in three-dimensional balls and their covering spaces. Based on work to appear with C. Keller, H. Ooguri, and I. Zadeh. 

One of the main characterisations of word-hyperbolic groups is that they are the groups with a linear isoperimetric function. That is, for a compact 2-complex X, the hyperbolicity of its fundamental group is equivalent to the existence of a linear isoperimetric function for disc diagrams D -->X.
It is likewise known that hyperbolic groups have a linear annular isoperimetric function and a linear homological isoperimetric function. I will talk about these isoperimetric functions, and about a (previously unexplored)  generalisation to all homotopy types of surface diagrams. This is joint work with Dani Wise.