Paul Horn

My research interests are in combinatorics. Specifically, I focus on the use of probabilistic, spectral, and geometric methods to understand graphs, which are a mathematical abstraction of networks. My particular work includes using ideas from 'continuous' mathematics, for instance the 'diffusion of heat' on a network, to understand structural properties of networks, for instance whether there are 'bottlenecks' in the network. I also use structural information about networks to understand random processes on graphs.

One recent project I've been involved in is the development of notions of 'curvature' for graphs. Curvature, in the study of manifolds (which are objects that 'locally' look like Euclidean space, in the way the surface of the earth 'locally looks like a plane), is a measure of how a space expands 'locally.' The notion of curvature we introduced has allowed me and my collaborators to prove graph theoretical analogues of a number of results from Riemannian geometry; specifically a graph theoretical version of the 'Li-Yau inequality' along with many consequences.