This short article is intended to be read by non-mathematicians who don't quite remember what the term "matrix" or "polynomial" refers to. I'll try to give you an intuitive idea of what a PhD student working on "Algebraic Topology" studies nonetheless.

First of all, studying or researching mathematics is not about remembering formula or calculating some really large numbers. That is a part of mathematics, but it's not what drives it, it is merely a tool that is more and more handed to computer systems.
So, what is mathematics instead? I am not competent to give an answer (and there have been many many different answers to that question in the past) but I can explain you my view on pure mathematics:

Pure Mathematics is the study (by any means) of the statements of which truth can be derived syntactically, i.e. by some computational process. (Yes, I'm quite a syntactic thinker).

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