Mathematicians need to talk and write about their mathematics.  This includes undergraduates and MSc students, who might be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  It can be helpful to think of this as a form of storytelling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.  In this interactive session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.

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.