My favorite kind of math questions are about structure and classification: How can we tell when two objects are the same? What are the intrinsic properties which differentiate them? 

This kind of thinking is apparent in mathematics from very early on: you can tell that a square and a triangle are not the same, and you can also tell apart a square from some rectangles, and (importantly) the properties that let you tell these two apart are different! I work on problems like this, where the objects are certain collections of functions called operator algebras.

For addressing such questions in my research, there are two main approaches. One is "bottom-up," and works with well-understood finite-dimensional approximations (random matrix models) in order to understand a large infinite object. The other one is abstract and "top-down," where you work with certain big invariants (the first-order logical theory) for the objects you want to understand. Both have their advantages, and I have a really fun time trying to fit them together. 

You can check out the more technical stuff via my papers below.