# Research

My research involves interactions between partial differential equations (PDE), numerical schemes, applied probability, and computer science. I am interested in both the rigorous analysis of PDE, and the development and implementation of algorithms. It is exciting when mathematical analysis and insights can lead to more efficient algorithms.

My current research is focused on nonlinear partial differential equation continuum limits for discrete sorting problems and applications in engineering and scientific contexts. This research is supported by the National Science Foundation under grants DMS-1713691 and DMS-1500829. Below are brief descriptions of some of my current/ongoing research, and past research. Note this is quite old and will be updated soon.

Nondominated sorting is a combinatorial sorting algorithm that is widely used for numerically solving multi-objective optimization problems. We established a Hamilton-Jacobi equation continuum limit for nondominated sorting and are currently pursuing applications to machine learning problems.

Our Hamilton-Jacobi equation continuum limit for nondominated sorting can be viewed as a stochastic homogenization result. Our original proof relied heavily on a continuum variational problem, and we have recently discovered a new proof using only PDE techniques. Part of my current research is focused on using this new proof to establish continuum limits for other important combinatorial problems.

Directed last passage percolation is an important stochastic growth model in statistical physics. It is closely related to directed polymers, the totally asymmetric simple exclusion process (TASEP) and the corner growth model. In this work, we proved a Hamilton-Jacobi equation continuum limit for the time constant in a directed last passage percolation model with discontinuous macroscopic inhomogeneities.

The representation of curves by integral invariant signatures is an important step in shape recognition and classification. For some of the most commonly used invariants, it is currently unknown whether the signature uniquely determines the shape. In this work we provide some uniqueness results for the commonly used circular area signature.

Image diffusion and sharpening are important problems with many applications (e.g., medical imaging, satellite imagery). In this work we obtained new and effective PDE-based image sharpening algorithms by recasting traditional gradient flow algorithms under the Sobolev gradient framework.