My research interests include geometric analysis, differential geometry, geometric measure theory, and partial differential equations. The following are some of the projects I am working on.
In large experiments or in large data sets collected for machine learning, the data involved often satisfies some nonlinear constraints, and lies on a submanifold of the space of (naively) all possible outcomes. To recognize patterns and structure in such data sets, some modern methods use the eigenfunctions of the Laplacian on graph approximations of the underlying manifolds. I am interested in why these methods work, and study embeddings of Riemannian manifolds by heat kernels and eigenfunctions of the Laplace operator. I showed that the number of eigenfunctions or heat kernels needed can be bounded in terms of geometric information. (arXiv:1311.7568 [math.DG])
When two manifolds are close, are the spectra of the Laplace operator close as well? This certainly depends on the chosen distance. Earlier work by Fukaya and by Cheeger and Colding shows that under metric measure convergence of the manifolds and with some curvature conditions, eigenvalues are continuous. I looked at the behavior under flat and intrinsic flat convergence. Without assuming curvature bounds, and with an assumption of convergence of the total volume, I could show semicontinuity of eigenvalues under flat and intrinsic flat convergence. (arXiv:1209.4373 [math.DG] and arXiv:1401.5017 [math.DG] respectively)
In an on-going project I study the behavior of a viscous drop of fluid surrounded by a second fluid, with surface tension on the separating interface. When one approaches this problem with techniques from geometric measure theory, it has similarities with a numerical method for dealing with fluid-structure interaction problems, the Immersed-Boundary Method.