Did This German Retiree Solve A Decades-Old Conjecture?

And then this came across my desktop (my iPad’s too old to work with the Twitter client anymore):

Retired German man solves one of world's most complex maths problem with 'simple proof' | The Independent https://t.co/eaD56anmRu

— Egan J Chernoff (@MatthewMaddux) April 3, 2017

The underlying news is that one Thomas Royen, a Frankfurt (Germany)-area retiree, seems to have proven the Gaussian Correlation Inequality. It wasn’t a conjecture that sounded familiar to me, but the sidebar (on the Quanta Magazine article to which I’ve linked there) explains it and reminds me that I had heard about it somewhere or other. It’s about random variables. That is, things that can take on one of a set of different values. If you think of them as the measurements of something that’s basically consistent but never homogenous you’re doing well.

Suppose you have two random variables, two things that can be measured. There’s a probability the first variable is in a particular range, greater than some minimum and less than some maximum. There’s a probability the second variable is in some other particular range. What’s the probability that both variables are simultaneously in these particular ranges? This is easy to answer for some specific cases. For example if the two variables have nothing to do with each other then everybody who’s taken a probability class knows. The probability of both variables being in their ranges is the probability the first is in its range times the probability the second is in its range. The challenge is telling whether it’s always true, whether the variables are related to each other or not. Or telling when it’s true if it isn’t always.

The article (and pop reporting on this) is largely about how the proof has gone unnoticed. There’s some interesting social dynamics going on there. Royen published in an obscure-for-the-field journal, one he was an editor for; this makes it look dodgy, at least. And the conjecture’s drawn “proofs” that were just wrong; this discourages people from looking for obscurely-published proofs.

Some of the articles I’ve seen on this make Royen out to be an amateur. And I suppose there is a bias against amateurs in professional mathematics. There is in every field. It’s true that mathematics doesn’t require professional training the way that, say, putting out oil rig fires does. Anyone capable of thinking through an argument rigorously is capable of doing important original work. But there are a lot of tricks to thinking an argument through that are important, and I’d bet on the person with training.

In any case, Royen isn’t a newcomer to the field who just heard of an interesting puzzle. He’d been a statistician, first for a pharmaceutical company and then for a technical university. He may not have a position or tie to a mathematics department or a research organization but he’s someone who would know roughly what to do.

So did he do it? I don’t know; I’m not versed enough in the field to say. It’s interesting to see if he has.

What I Learned Doing The A To Z Project

So now I’ve had the chance to rest a little and recover from the Summer 2015 Mathematics A To Z Project. I’d been inspired to it by Sue Archer and her Doorway Between Worlds blog. I had piggybacked on her discussing the word “into” with a description of its mathematical use.

The first thing I learned is that it’s easy to write several week’s worth of essays in a big session if I have a clear idea what they’re all to be about. That left me feeling good. I do worry when I go several days without anything fresh or more than the reblog of someone’s interesting pictures. I like sharing someone else’s interesting pictures too, mind you. I just know it’s not work on my part to share them. Also when I had to travel a while in May and June, and when my computer was out for repairs, I didn’t have to scramble to do anything.

Another is that I liked the format, which had me jumping around several concentrations of mathematics. It also had me jump from common enough levels into math-major stuff all the way to grad school stuff. I particularly liked trying to introduce graduate-level mathematics in tolerably clear English and in around a thousand words. Helping me out here was the Hemmingway Editor, which attempts to judge how complicated one’s writing is. It’s in favor of shorter, clearer sentences with fewer adverbs and no uses of the word “very”. I can’t agree with everything it judges. It’s a computer, after all. But writing about advanced subjects while watching how complicated my sentences came out has helped my prose style.

Something else I’ve learned from this is that there’s a taste for pop-mathematics about more advanced topics. It’s easy to suppose that people who never studied, or never liked studying, mathematics are most likely to read about the easy stuff. That’s probably not quite so. Probably what people really want is to feel like they’re being let in on the cool stuff. Mathematics has a lot of cool stuff. A lot of it requires a long running start, though. For example, I couldn’t talk about a ring until I’d described what a group was. So that essay felt like it was taking forever to get started while I wrote it. I don’t know how it felt to people reading it. The z-transform, similarly, has a lot that’s neat about it, but it took a while to get there. I hope it stayed promising long enough for people to stick through it.

My terror throughout writing all 26 entries was that I was about to say something really, obviously stupid, and that a flock of mathematicians would descend on me. Scorn and semiprofessional humiliation would follow. I’m still a bit worried, although nobody’s said anything too bad to me yet.

The project was quite good for my readership. Between the A to Z essays, Reading the Comics posts, occasional other essays, and reblogs, I went a solid thirty days with something new posted every day. That’s surely why June was my most-read month here ever. And why July, though having fewer posts, was still pretty well-read. I confess I’m disappointed and a bit surprised I never hit the “Freshly Pressed” lottery with any of these. But that’s just the natural envy any writer has. Everybody else always seems to be more successful.

I’d like to do a similar thematically linked project. I might in a few months do another A to Z. I’m open to other themes, though, and would love to hear suggestions.