Mathematicians, and to a lesser extent academics, represent one of the most misunderstood professions. As a result, these pages present some of my expanded thoughts on what being a mathematician and an academic means.
You're a mathematician? What do you actually study?
I get asked this on occasion. Although mathematicians know well what they actually do, this remains a tough question to answer. The difficulty arises from the dilemma that the question has either a five-second answer (which is too short to be helpful) or a fifteen-minute answer (which tries the audience's patience).
However, the answers are important, in particular since such a wide gulf has developed between what is taught in mathematics curricula and what mathematicians actually think about. Neither mathematicians themselves nor the education system have been particularly successful at bridging that gap. Nowadays, this gap is often so absurdly wide that when university students progress from basic lower-level courses (e.g., calculus, differential equations, and linear algebra) into the rigorous advanced courses (e.g., real and complex analysis, abstract algebra), they basically must learn a whole new field and pick up an entirely new way of thinking.
These pages are an attempt to give "fifteen-minute answers" to the question—"What do mathematicians actually study?"—and to other tangentially related questions. Of course, this comes with a disclaimer: what is written is merely my own views and are not definitive, and other mathematicians may have some different perspectives.
You're an academic? What do you do all day?
Very generally, the job of an academic and researcher is to do the following: