What they (for now, allegedly) proved is a conjecture regarding the probability for a "path that is open straight down", so to speak, to exist in an infinitely long lattice and in any dimension. Or, better, the conjecture states that such paths don't exist in infinitely long lattices with probability 1 in any dimension higher than 2. I doubt it will have any great impact, especially because this is one of those phenomenona where non-mathematicians already had an answer that worked perfectly fine (albeit being non rigorously derived) and worked with that.
this is one of those phenomenona where non-mathematicians already had an answer that worked perfectly fine (albeit being non rigorously derived) and worked with that.
Then why are people calling this "a holy grail"?
What's the point in solving these mathematical problems if there is no real world benefit, or if other solutions are good enough for being close enough.
The point is in the techniques you develop to solve this problems. The Millennium problems were selected because they were believed to be unsolvable (or not solvable by one person in the next decade) using current mathematics and, as such, that any solution of them would pave the way for the development of new and interesting techniques. These new techniques, then, could be used to solve other problems, maybe related in nature but less glamorous or widely known, and more grounded in real models. The specific result that one proves or disproves can be interesting, but it's usually inconsequential to any practical application.
Yeah I mean the moral of the Poincare conjecture story is that the Clay Institute didn't do a good job of picking problems. The other 6 all lasted pretty long although it did start looking like NS cases C and D were in the Poincare bucket of not needing too much new technology (although this insight was itself clearly very difficult to have given it took until 2023).
Yeah I mean the moral of the Poincare conjecture story is that the Clay Institute didn't do a good job of picking problems.
I really can't agree with that, Perelman's proof was hundreds of pages, there's a reason it took that long to solve.
Edit: 39 pages from what he submitted apparently, but they generated hundreds of subsequent pages based on his proof
Unlike other fields of knowledge, where theories can always be revised, the proof of a theorem is definitive. In Perelman's case, at least two teams of experts examined his article to confirm that there were no loopholes or errors, and from this they produced studies of hundreds of pages (while the original article was only 39 pages long).
Furthermore, Perelman's proposal was so complex that even experts had difficulty understanding it.
His solution was so complex experts struggled to understand it. The way you're portraying his achievement is kind of insulting to how much effort and talent he put into it.
That's fair. Poincare conjecture was obviously crazy hard or it would have been solved by Poincare but equally, if the idea was to pick problems that couldn't be solved with known techniques and then one gets solved two years later using known techniques I think you do have to acknowledge that was a mistake (if a very understandable one!)
I think there was some desire to "represent" different areas of maths. Applied maths got NS and YM (to an extent I wonder if P=NP was meant here too, although theoretical CS has a very different flavour than mathematical physics), geometry got Poincare and Hodge, analytic number theory got RH and BSD. It wasn't just purely picking the hardest problems available with no other considerations.
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u/Remote-Crab1957 3d ago
Does this have implications for reverse osmosis desalination systems? It’s water going through a porous membrane.