I am a first-year PhD in Mathematics at UCLA. I graduated from Rutgers University in 2015. Right now I'm interested in harmonic analysis and PDEs. I also like functional analysis and topology. But I've got at least a passing interest in almost everything else!
My upcoming reading list, once I find more free time:
The Analysis of Linear Partial Differential Operators, Lars Hormander.
Ordinary Differential Equations, and Mathematical Methods of Classical Physics, V. I. Arnold.
Elliptic Partial Differential Equations of Second Order, Gilbarg & Trudinger.
Singular Integrals and Differentiability Properties of Functions, Elias Stein.
Introduction to Harmonic Analysis, Yitzhak Katznelson.
Recently I've become interested in the geometrical structure of solution spaces to differential equations. Some digging around led me to the papers of Vinogradov and the ideas of jet spaces, secondary calculus and diffieties. I think I need to bring my algebra and geometry game up before I proceed, but I'd really like to see what lies down this road in the less-near future.