NewScientist (5 August 2020, paywall) interviews post-doc mathematician Lisa Piccirillo, who solved the Conway knot question concerning slicing in a week, concerning why she became a mathematician, and what it takes:
The decision to go to graduate school was a difficult one. I still had this idea that I think a lot of people have, which is that the only way to be a successful mathematician is to be a genius, and I’m certainly not anything like that. So I thought: “Why bother? I’m never going to be that good.”
There’s a strong stereotype of what people who do maths are like – introverted, nerdy, probably male, probably dead – and I was none of those things. I was very worried that I would have to give up other aspects of myself to be a maths robot and I didn’t want to do that. I felt that tension very acutely in my undergraduate programme, but in graduate school, I learned that this tension isn’t real. Mathematicians are interesting humans and none of them are geniuses.
Oh, ouch. I’ll bet there were some hurt feelings over that one. But Piccirillo has her revenge on me just for writing this post:
NewScientist: What will you be working on next?
I’m still very interested in 4-manifolds and in using sliceness to understand them better. It’s also true that this trick I used for the Conway knot doesn’t work on some other, more complicated knots. The reason is because it isn’t always possible to build a trace – sometimes it’s provably impossible or we just don’t know how to do it.
I’m trying to understand how to apply this type of argument more broadly to sliceness problems. More concretely, it turns out that sometimes, for some special knots, I can go home and build you another knot that shows a trace, but a computer can’t. Why not? It’s because we don’t know the rules of how we do it ourselves. If the maths gods hand me a knot and ask me to build a trace, I may get lucky, but I don’t know if I could tell you how I got there. And I’d like to understand why.
Concerning a trace: All knots have something called a trace, which is the manifold you can build from that knot. And a manifold?
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or n-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension n. [Wikipedia]
OK, that’s just a digression. My actual interest is in her statement … If the maths gods hand me a knot and ask me to build a trace, I may get lucky, but I don’t know if I could tell you how I got there. That just leaves me hanging, being a software engineer and all. She doesn’t know how? What? Then how does she know the trace properly derives from the knot? Given a trace, is there a trapdoor function that reveals the knot to which it corresponds?
Augh!
