https://arxiv.org/pdf/2004.06929.pdf

https://arxiv.org/abs/2003.02844

https://arxiv.org/abs/2003.10466

https://arxiv.org/abs/2004.00015

https://arxiv.org/abs/2004.06115

https://arxiv.org/abs/2003.02860

A motivation in the development of string theory was the 'duality' flip, exchanging the s- and t-channels, which relates all the cubic Feynman graphs at each order in perturbation theory, with fixed planar structure. In string theory, we can understand this as coming from the moduli spaces of marked surfaces, with the cubic diagrams corresponding to complete triangulations. I will describe how geometric-type cluster algebras give a surprising 'linear' way to talk about the same combinatorial problem, using results from work with N Arkani-Hamed and H Thomas and G Salvatori. This gives new ways to compute cubic scalar amplitudes, and new families of integrals generalizing the Veneziano amplitude.

In this talk I will present results from an ongoing joint research  program with Konrad Waldorf. Its main goal is to understand the  relation between gerbes on a manifold M and open-closed smooth field  theories on M. Gerbes can be viewed as categorified line bundles, and  we will see how gerbes with connections on M and their sections give  rise to smooth open-closed field theories on M. If time permits, we  will see that the field theories arising in this way have several characteristic properties, such as invariance under thin homotopies,  and that they carry positive reflection structures. From a physical  perspective, ourconstruction formalises the WZW amplitude as part of  a smooth bordism-type field theory.

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

One of the main concerns in social network science is the study of positions and roles. By "position" social scientists usually mean a collection of actors who have similar ties to other actors, while a "role" is a specific pattern of ties among actors or positions. Since the 1970s a lot of research has been done to develop these concepts in a rigorous way. An open question in the field is whether it is possible to perform role and positional analysis simultaneously. In joint work in progress with Mason Porter we explore this question by proposing a framework that relies on the principle of functoriality in category theory. In this talk I will introduce role and positional analysis, present some well-studied examples from social network science, and what new insights this framework might give us.