I'm so excited in anticipation of my near-term return to studying math, as an independent curiosity hobby. It's going to be epically fun this time around with LLM's to lean on. Coincidentally like Terence Tao, I've also been asking complex analysis queries* of LLM's, things I was trying to understand better in my working through textbooks. Their ability to interpret open-ended math questions, and quickly find
distant conceptual links that are helpful and relevant, astonishes me. Fields laureate Professor Tao (naturally) looks
down on the current crop of mathematics LLM—"not completely incompetent graduate student..."—but at my current ability level that just means looking
up.
*(I remember a specific impressive example from 6 months ago: I asked if certain definitions could be relaxed to allow complex analysis on a non-orientable manifold, like a Klein bottle, something I spent a lot of time puzzling over, and an LLM instantly figured out it would make the Cauchy-Riemann equations globally inconsistent. (In a sense the arbitrary sign convention in CR defines an orientation on a manifold: reversing manifold orientation is the same as swapping i with -i. I understand this now, solely because an LLM suggested looking at it). Of course, I'm sure this isn't original LLM thinking—the math's certainly written down somewhere in its training material, in some highly specific postgraduate textbook I have no knowledge of. That's not relevant to me. For me, it's absolutely impossible to answer this type of question, where I have very little idea where to start, without either an LLM or a PhD-level domain specialist. There is no other tool that can make this kind of semantic-level search accessible to me. I'm very carefully thinking how best to make use of such an, incredibly powerful but alien, tool...)