Describing my research to non-mathematicians is tricky. The precise statement of my theorem usually requires a lot of background that even mathematicians in different fields may not have; much as I love explaining maths to people, it’s just not practical to expect non-mathematicians to absorb a bunch of definitions well enough to understand what I’ve done with them. Instead, I try to give people the general picture of what I’m studying.
It’s much harder to say something vague but not inaccurate than to say something precise. At the beginning of a project I haven’t yet developed my own intuition, and I don’t have a good sense of what’s essential and what’s just a minor detail. Figuring out how to explain my project concisely to non-mathematicians is part of the process of really understanding what I’m trying to do and why. (It also allows me to answer the question “So what’s your research about?” with something that will satisfy those who only asked to be polite and inspire further questions in the genuinely curious.)
After many months, I’ve come up with the following summary of my latest theorem:
These shapes we’re studying aren’t as twisty as one might have thought.