My answer to a traditional question: “You are a mathematician. What do you actually do?”
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 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 film.