I have compiled a list of things to say and not to say to a mathematician while making small-talk. (This list is especially relevant to graduate students in math.)
Good: So, what are you researching?
Bad: Do you have an adviser picked out yet?
In the event that the mathematician you’re talking to is a first or second year and doesn’t have an adviser, the question at least allows them the ‘out’ of saying, “No, I’m busy studying,” or “No, but I’m looking for an adviser!” The latter pushes the issue, and might make your conversation partner squeamish.
Good: Who’s been your favorite professor?
Bad: What’s been your least favorite course?
Let’s face it, most people like to focus on the positives as opposed to the negatives, anyway. But thinking about some courses can cause symptoms akin to PTSD. (I still have undergraduate analysis blocked out of my memory.)
Good: Are you in the applied program, or pure?
Bad: Do you think your program is better than the other?
It’s no secret that pure mathematicians are a little arrogant over their rarefied status in the mathematical community. On the other hand, the applied folks can become a bit smug over the dollars they rake in. There’s a fair amount of dissention, and there’s no need to stir it up.
Good: What are you teaching?
Bad: What are you teaching?
Teaching can be both good and bad, just as this question. Use at your own risk.
Good: Where did you do your undergrad?
Bad: Are you a constructivist?
There’s strong opinions on both sides of the aisle when it comes to constructivism. Best to steer clear of that conversation.
Good: Where are you originally from?
Bad: How do you feel about the axiom of choice?
Most mathematicians accept the axiom of choice without too much thought. But you don’t want to end up asking this to one of those weirdos who doesn’t accept it, trust me.
Bad: Don’t you just love cohomology memes?!
This should need no explanation.
Good: What do you like to do in your free time?
Bad: Is the real part of every non-trivial zero of the Riemann zeta function ?
Perhaps the mathematician you’re talking to likes to spend their free time thinking about the Riemann hypothesis. But this is again one of those controversial topics which would be better to stay away from.
Good: (After you learn Rouche’s Theorem:) Want to see a proof of the fundamental theorem of algebra?
Bad: (Same as above, except you pose the question at an inappropriate time.)
Did your mother teach you any manners?
Bad: Didn’t Wiles originally have some errors in his proof of Fermat’s Last Theorem?
Never. Disrespect. Sir. Wiles.
I hope that gets you started! -AJB