### Abstract

I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the Redwood Center for Theoretical Neuroscience and mathematicians at U.C. Berkeley. I am also interested in theoretical questions involving semidefinite programming, optimization, and computational algebra. The following is a description of several interrelated lines of research in which I will actively participate in the coming years. The first three sections contain very brief discussions of topics related to theoretical neuroscience that I have only begun exploring in recent months. The final sections describe more theoretical studies that I have been investigating in recent years and therefore contain more detailed descriptions. 1. Sparse coding and compressed sensing Sparse coding refers to the process of representing a real vector input (such as an image) as a sparse linear combination of an overcomplete set of vectors (called a sparse basis). Here, overcomplete refers to the fact that there are many more vectors in the sparse basis