Most of my tutors spend their time working with large systems of linear equations, which is just applied maths, normally specifically for applications within biology or chemistry. They tackle problems such as "how do you store a sparse 1,000,000x1,000,000 matrix in a computer", "how can you 'solve' these matrices using a method which does not fill in the sparse matrix, and therefore make it too big for any modern computer to store", etc.
The other half of my tutors seem to work in quantum research.
Another works in ultra efficiency. How can you design code and hardware that is ultra, ultra efficient, so you can send probes far out in to space or to the depths of the ocean with the least amount of energy.