What do pure mathematicians do?

My answer to a question in Quora: What do pure mathematicians do?

I have already answered this question elsewhere: what do I do as a pure mathematician? I study symmetries. More precisely, I study multi-mirror symmetries, where mirrors get reflected in other mirrors, and reflections breed and multiply all over the place. I design ways to find safe paths in the maze of mirrors, and to tell real objects from sneering phantoms.

If you happen to be involved in a shoot-out in a Hall of Mirrors — like in the famous scene from Orson Wells’ The Lady from Shanghai, see a YouTube clip:

— I can provide a consultancy, at UCU recommended professorial rates.

This clip also gives an answer to a question I heard from a graduate student: why all calculations in simple algebraic groups end up in manipulations with root systems? Because a root system is just a convenient way to write down on paper the skeleton mirror system of the group, and calculation with roots is the final shoot-out at the end of the theorem.