I develop numerical methods to study PDEs on metric graphs. Because of the coupling conditions at the vertices, standard one-sided finite difference methods for the junction conditions reduce the accuracy below the local truncation error of the method. Some alternative methods include the addition of ghost points, the discontinuous Galerkin method, or spectral methods. 

The spectra of the eigenvalue problem on the metric graph provides insights into the structure of the graph itself. The eigensolutions form a complete orthogonal basis for the graph so they can be used to write the solutions to other linear PDEs. Some eigensolutions are nonzero on only a small subset of edges (i.e. they are localized). We can derive conditions for the lengths to ensure exact localization. We can also prove that some shapes can never be fully localized.

I have developed a new mathematical tool in which to study the spatial spread of epidemics. There is extensive evidence that transportation systems such as air, rail, rivers, and roads may enhance the spread of an epidemic into the surrounding geographic regions. This novel mathematical model couples ODEs at the vertices with the junction conditions on the metric graph. The solution on the edges of the metric graph, in turn, is coupled to the 2D continuum as a boundary condition.

The wave equation on a metric graph is a good way to model a new type of laser called a "random network laser." These lasers are comprised of a series of interconnected optically active fiber waveguides. They work by using multiple scattering (usually a feature that lowers laser performance) as a source of optical gain. Different frequencies of light occupy different areas in the network (a phenonemon akin to localization). Following the framework of Gaio, et al. we use a Buffon's needle graph to model such a laser.

I have worked with experts in fields from public health to number theory. I find great value in cross-disciplinary collaborations.

Here are some phenomena that are enhanced by network structure.  There are lots of ways we can apply PDEs on metric graphs to "real world" problems.

• Large mammals have been shown to travel along narrow geographic corridors like roads, rivers, and pipelines. (McKenzie 2012, Serrouya 2020, Dickie 2020, Mumma 2019, Newton 2017)

• Insects may migrate along roads, hedgerows, and other corridor-like geographic features. (Zamberletti 2021, Roques 2016)

• Prehistoric human migration and settlement may have been influenced by the path of rivers. (Coulthard 2013, Twilley 2016, Ollivier 2016, Kidder 2008)

• Trade routes may have accelerated the adoption of both Islam throughout Asia and northern Africa and Christianity through southeast Asia and parts of Africa. (Michalopoulos 2018, Lovejoy 1971, Hill 2009, Munson1968, Harrak 2009, Seland 2012)