I promised a followup and unlike those safe-opening crackpots, I deliver. Brief summary of parts I and II in this paragraph. A few physics collaborators and I stumbled across an interesting linear algebra formula that relates eigenvectors and eigenvalues. It seemed so simple we thought for sure it must be known in the literature, but couldn’t find anything. After posting here, you guys directed me to Terry Tao who promptly replied to our email with three proofs.
After barely managing to process one proof, we decided to go for and see if he’d like to write up a paper. I sketched up a draft figuring if we had something that already looked good he’d be more likely to say yes. He promptly replied and said sure (I screamed a little bit), offered a corollary and a few other neat observations. At this point I was two proofs, a corollary, and some other new things behind. I hacked my way through the new information and was about to send a v2 of the draft the next day when he sends another proof (now I’m three proofs behind, oof, I seriously wondered how I would ever catch up with this). At some point during this story, a colleague of mine who straddles physics and math said, “He’s famously like a cheery firehose of mathematics, Guess he’s power-washing you today.” I felt clean.
Anyway, I finally caught up and the firehose slowed down a bit. We put the paper online last weekend and it finally appeared on the arXiv, along with a new Terry blog post! I’m so excited you guys don’t even know.
As for the math, the arXiv paper is barely over two pages so you’re best off reading it there or on his blog rather me trying to write formulas here on reddit. Also, as I was typsetting Terry’s proofs, I had two files going, one called Math.tex (that ended up being the paper) and another called Physics.tex. The former was basically just what he had sent us slightly reformatted with a few additional notes. The latter described the first proof in enough detail such that I or my physics collaborators could understand it. The latter is about five times as long as the former, heh.
Terry has been a pleasure to work with; I learned a ton and he seemed really chill whenever I would say things like, “I have no idea how this normally works in math but…”
In other news, my Erdos number just went from 4 to 3 where it will probably remain for the rest of my life.