“The essence of Mathematics lies in its freedom” -- Georg Cantor

I like to sit in the boundary between pure and applied mathematics, although with a tilt towards the pure side. I am interested in pure math questions that arise from physics problems. More precisely, I like taking a real physical model and tweaking something so that it is not "physical" in the sense that it no longer applies to reality. However, we can still ask question about this new model to see how it behaves. My favorite example is if you take the universe as it currently exists, but you change gravity to be a repelling force instead of attracting. We can take this new model (that isn't physical because it doesn't describe our reality, but is a perfectly reasonable alternative) and ask the same kinds of questions you would about our universe. Can you still make atoms, planets, stars or galaxies? If not, what kind of structures do exist in this new universe?

Lately I have been looking into fractal geometry. I've always been fascinated by fractals and their applications to nature's geometry. Many things in our universe obey some self similarity scaling model (moons move around planets, planets move around a sun, suns move around a galaxy), but Euclidean geometry doesn't do a very good job of explaining this. Our buildings and houses don't look like rocks or mountains or trees, so there is a different geometry being applied in nature versus how humans have decided to build. I like to study how nature decides to build things.