Math

Research

I'm working with the intersection of large-graph limits, variational calculus, and nonlocal analysis. My current research involves graph Ginzburg-Landau theory and nonlocal perimeter. This research has applications in graph cut problems, image segmentation and social network analysis, among other questions. This research is in collaboration with James Scott, Qiang Du, and Mason Porter.

The next piece of this project is to consider dynamics on graphons, in particular the graphon heat equation as studied by Medvedev and the graphon Allen--Cahn equation as an extension of the work by Kaizheng Wang.

If you're interested in collaborating or learning more about these research projects, please email me!

CV

Publications

• Ginzburg--Landau (GL) theory, which was originally developed to model superconductors, has more recently been applied to a variety of applications including graph clustering, image processing, and min-cut/max-flow problems. These applications typically use the graph GL functional. Graphons are a notion of large-graph limit (and the notion of graph-to-graphon convergence is defined with respect to a norm called "cut norm"). This paper defines a graphon GL functional, and shows that the minimizers of the graph GL functional converge to the minimizers of the graphon GL functional.

• Variational inference is an approach to Bayesian inference that is based on an optimization problem. It is widely-used, but VI algorithms have lacked theoretical guarantees. We express variational inference as an approximate Wasserstein gradient flow, and we discuss theoretical implications.

• Mode-connectivity is an interesting property that has been discovered in neural net loss functions, which are high-dimensional and nonconvex. It has been related to a "no bad local minima" property that's been noticed in practice, and is related to overparametrization. This workshop paper demonstrates and begins to explain the existence of mode-connectivity in the ELBO, which is the loss function of variational inference.

Teaching

Seminar