Research

Stability in the KPZ universality class

Advsior: Dr. Firas Rassoul-Agha

Random geometry models such as Last Passage Percolation (LPP), Brownian LPP, and the Directed Landscape can be viewed as dynamical systems with semi-infinite geodesics acting as energy minimizing paths. In all of these models, there are directions in which there are two distinct geodesics and thus the dynamical system is unstable. I study these instability graphs and their relationships to shocks in the various models.

Dilation MRA and distribution learning

Advisor: Dr. Anna Little

If a signal is corrupted with noise and then randomly shifted and dilated, can it still be recovered? This project expands on the field of Multi-Reference Alignment (MRA) by incorporating random dilation as well as random translation and noise. Recovery is possible via leveraging wavelet invariants to map dilations into translations.

Another take on the MRA problem is Moment Invariant Distribution Learning (MIDL), where a translation distribution is sampled and a parent distribution is randomly shifted by these samples into many child distributions, from which very sparse samples are taken. If enough samples of the translated parent distribution are taken, can the parent distribution be recovered, even under sparse sampling of each child distribution?

Finally, I am interested in the continuum MRA problem, where the signal and the translation distribution are taken to be fully continous.

Random convex hulls

It is known what the expected length and area are for the random convex hull formed by a Brownian motion up until time T…

but what about the expected area and length for the convex hull of a Brownian motion run up until the first time it exits the unit circle?

Turns out stopping times don’t play nice with rotatational symmetry and dimension reduction. This is an ongoing work in collaboration with Samantha Linn and Tory Richardson.