Research

Stochastic PDE’s and random geometry

Advsior: Dr. Firas Rassoul-Agha

I am interested in the study of stochastic Hamilton-Jacobi (SHJ) equations via the behavior of their characteristics. From the perspective of random dynamical systems, these characteristics can be considered to be semi-infinite geodesics in random geometries of the KPZ universality class. Thus, the study of geodesics in random geometry models such as Last Passage Percolation (LPP), Brownian LPP, and the Directed Landscape is integral to understanding how stochastic PDE’s evolve in time. Specifically, I am interested in the interplay between instability and shocks in such SHJ equations.

Data science over groups, Cryo-EM

Advisor: Dr. Anna Little

I am also interested in a wide range of inverse problems concerning recovering a signal from noisy samples perturbed by a group action, such as rotation or translation. A motivating biological problem for this field is the Cryo-EM problem, in which many (possibly) identical molecules are frozen in a solution and are imaged all at once in a single frame. The signal to noise ratio is generally very small, and one has to leverage the large number of samples of the molecule that one has. However, each sample is rotated in 3D and then tomographically projected down into the plane. While this problem is notably difficult, there are still rather difficult but tractable toy problems such as Multi-Reference Alignment (MRA) and Multi-Target Detection. Of particular interest to me is the construction of estimators for such problems with provable convergence rates in sample size and theoretical guarantees for recovery. Additionally, I enjoy working in the full continuum.