Generally discovering any kind of new math basically boils down to asking some question and hunting for an answer. A fun example might be something like the [Millennium Prize Problems](https://en.wikipedia.org/wiki/Millennium_Prize_Problems). This is a set of unsolved problems in math that each have a million dollar prize to anyone who can solve them. The Navier-Stokes problem, for example, is actually a really straightforward equation to write down if you know some basic physics, but it’s a differential equation, which isn’t necessarily very useful for many practical applications. And so the million dollar question is basically whether or not we can take this impractical equation and rewrite it as a practical one instead. [Numberphile did a video](https://www.youtube.com/watch?v=ERBVFcutl3M) on this topic if you’re interested.

Some answers about what mathematicians do in general, but here’s some more detail about day to day.

First of all, there’s all the non-research stuff: teaching and prep, meetings, mentoring grad students, etc.

But for actual research: reading papers (keeping up with what’s new in the field and looking for a gap in existing research to work on), working out examples by hand or with code to find interesting questions and form conjectures, using those examples to build intuition and work towards a proof (often starting with a simple case and then working to generalize and strengthen the result), writing out the proof and working out details.

Mathematicians may be working on several projects at once in different stages. Some work alone, some do a lot of collaboration and bouncing ideas off of each other.

Sometimes, people state a theory that they haven’t proven – most famously, Fermat’s Last Theorem. They might work on this to solve it.

There are “problems” in which we can predict what the answer can be, but we haven’t conclusively proven it yet.

They might also take problems or demonstrations from more [practical examples](https://youtu.be/rXfKWIZQIo4?si=x-1fb13y8xBE5_x1), or applied fields like physics, and work out the maths for them.

There’ll also be a lot of teaching, marking, writing, supervision, grant applications etc. going on.

I really like the channel [Numberphile](https://youtu.be/d8TRcZklX_Q?si=GwQwk_C3wIBbYwMx) if you want to find out more about both serious and more whimsical stuff – a lot of it just seems “ooh cool!”, but may have practical applications in the future.

Understand the difference between what I’ll call problems and exercises.

Exercises are what you do in math class when you are learning. The method for solving them is known and you just have to follow the right steps to get the answer. This is most people’s experience of math so they tend to envision mathematicians sitting around all day doing problem sets. Mathematicians do not do this. This would be like an author spending their time spelling random words.

A true problem is one where you don’t know how to solve it when you start. Finding a solution takes creativity and hard work. Sometimes years. The frontiers of math where the cutting edge research is happening are often so advanced that it might take years of study just to understand what the problem is (consider [this list](https://en.m.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics) of unsolved problems; I have a BS in math and it’s Greek to me).

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