2320: Millennium Problems

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Millennium Problems

Title text: The hard part about opening a hole in the proof of the Poincaré conjecture is that Grigori Perelman will come out of retirement to try to fix it by drawing a loop around the hole and contracting it to a point.

Explanation

Randall, drawn as Cueball, is presenting a slide on the Millennium Prize Problems, seven problems designated by the Clay Mathematics Institute in the year 2000 as some of the most important unsolved problems in mathematics, a sort of successor to David Hilbert's list of 23 problems announced in 1900. The seven problems are:

1. The P versus NP problem, the problem of whether or not a problem whose solutions can be verified in polynomial time must necessarily have a method for producing a solution in polynomial time. This is thought not to be the case, i.e. "P != NP", but is not proven (nor mentioned on Cueball's slide).
2. The Hodge conjecture in algebraic geometry.
3. The Poincaré conjecture, which asserts that the 3-sphere (the "surface" of a four-dimensional ball) is the only closed and simply-connected (i.e. no holes) 3-dimensional space. It was solved in 2003 by Grigori Perelman and is the only one of the seven problems solved to date.
4. The Riemann hypothesis, which asserts that all non-trivial zeroes of the Riemann zeta function have real part one-half.
5. The Yang–Mills existence and mass gap, the problem of why the color force is conveyed by massless gluons but observed only in massive particles. This one is not mentioned on Cueball's slide.
6. The Navier–Stokes existence and smoothness problem, which questions whether or not there must be a solution to the Navier-Stokes equations (the laws of fluid motion) for any smooth set of initial conditions.
7. The Birch and Swinnerton-Dyer conjecture, abbreviated "Birch/SD" here, which asserts that there is a simple way to tell the number of rational solutions to an elliptic curve.

There are $1,000,000 prizes attached to each problem, although Grigori Perelman, the mathematician who proved the Poincaré conjecture, turned down his prize.

Randall is attempting to demonstrate relationships between the various problems. According to the presentation, proving one might either disprove or prove others, and the proposed interactions between problems are so complex that the Institute might decide to award an additional prize to whoever can figure out which problem or problems have actually been solved by any given proof. This eighth prize could perhaps be funded by the award Perelman rejected.

Randall has previously been banned from conferences for various provocative acts; presumably he's on his way to getting thrown out of the Clay Mathematics Institute as well, as the "other" Cueball is already calling security. However, this seems to be only these three people, thus not a conference to be banned from this time.

The title text mentions that, if someone were to find a hole (a common expression for a deficiency or error) in Perelman's proof of the Poincaré conjecture, the famously reclusive author might show up again and fix the problem by applying the theoretical mathematics of differential geometry, where "hole" has a different meaning, to the figurative "hole" in the sequence of logical conclusions. The suggested method of enclosing the hole in a loop and then shrinking it away is reminiscent of the specific technique (Ricci flow with surgery) by which Perelman solved the Poincaré conjecture.

Transcript

[Randall, drawn as Cueball, is holding a hand palm up towards a screen where a projector on the floor in front of him is projecting a diagram. The projector is propped up on some kind of legs to project up on to the screen. Behind the projector Ponytail is watching him, while Cueball is looking away from Randall, while yelling after someone off-panel.]

Randall: ...Proving that one of these four is unsolvable, but not which. If it's one of these, it would open a hole in Perlman's Poincaré conjecture proof.

Randall: But it would also mean that solving either of the other two would re-prove Poincaré, and imply Hodge is isomorphic to...

Cueball: Security?!

[The slide on the projector screen shows a four-by-four matrix with 16 illegible entries, and also illegible text left and right of the matrix. The matrix is connected by four lines to four text segments written around the matrix. Two above (left and right), one to the right (at the bottom) and one below to the left. Arrows go between the right and left text at the top and both from the top left and the right text to the text at the bottom. The two to the right are connected by a line with an illegible equation written over this line, intersecting it. From the bottom text below the matrix an arrow goes down to another text beneath it. And from there an arrow goes up to the right text.]

Hodge

Riemann

Navier-Stokes

Birch/SD

Poincaré wrong??

[Caption below panel:]

I'm trying to make it so the Clay Mathematics Institute has to offer an eighth prize to whoever figures out who their other prizes should go to.

Trivia

• Randall misspells Perelman as "Perlman" in the sentence in the panel but spells it correctly in the title-text.

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Discussion

Ironically, Randall misspells Perelman as "Perlman" in the comic but spells it correctly in the alt-text.

172.69.63.147 02:56, 16 June 2020 (UTC)

Perhaps he meant Perlman the Perl-programming superhero? ;) 162.158.123.145 03:33, 16 June 2020 (UTC)

Or perhaps Ron Perlman wrote his own proof on his spare time from acting but never published it? --108.162.229.234 13:00, 20 June 2020 (UTC)

Ironic perhaps, but at whose expense? ;-) --172.68.215.141 20:44, 17 June 2020 (UTC)

There has been some controversy over the millennium prizes, given that in mathematics important results are often a product of the work of different mathematicians who are not necessarily close associates. Perelman reportedly believed that his work was a corollary to prior work by Richard S. Hamilton.

I think the idea of this comic is an extension to a question, which I've seen before in this discussion, "what if person A shows that 2 millennium problems are equivalent, and then person B proves one of them?" Should person B get both prizes, or should person A get one of them? It is easy to think of situations where it is hard to know who deserves the credit, and I think this comic takes that to a logical exteme. Probably not Douglas Hofstadter (talk) 03:59, 16 June 2020 (UTC)

The Wikipedia article for Grigori Perelman states the following: "The Clay Institute subsequently used Perelman's prize money to fund the 'Poincaré Chair', a temporary position for young promising mathematicians at the Paris Institut Henri Poincaré.", so no funding would be available for Randall's eighth prize. 162.158.74.61 04:21, 16 June 2020 (UTC)

By process of elimination, the matrix and the equation should represent Yang-Mills and P=NP, but which is which? The 4x4 matrix could represent the 4D unitary transformation from Yang-Mills? The equation seems to say 'Ar + (squiggles)' but I can't think of any complexity problems that might take this form. --Quantum7 (talk) 06:35, 16 June 2020 (UTC)

Is "millennium problems" also a pun on "millennial problems", i.e. those issues which seem straightforward to adults but baffle the younger generation (the "millennials")? See for example comic 2165. --188.114.102.48 00:48, 17 June 2020 (UTC)

no, because it wasn't created by Randall, but by the Clay Mathematics Institute, and they weren't in a punny mood at the time. Also, the prize was created a few years before millennials became enough of a 'thing' for such a pun to be funny. And finally, I'm too old to be a millennial and I don't find proving any of these propositions straightforward at all, even the one for which a proof has already been found. 12:01, 20 August 2020 (UTC)

The image is projected by a projector on the ground that Cueball is apparently standing in the way of, but there's no Cueball-shaped shadow on the projected image. 108.162.219.192 (talk) (please sign your comments with ~~~~)

"there's no Cueball-shaped shadow on the projected image." - of course not! Cueball is clearly constructed from lines - which (of course) have no width and therefore zero area and as a consequence, cannot obstruct any photons to cause a shadow to form. 172.69.70.213 02:13, 17 June 2020 (UTC)

Ha ha. No I think it is easy to see that Randall/Cueball is actually standing to the side of the projectors beam and he is thus not in front of the projector; it is thus not strange that his shadow is not there! --Kynde (talk) 09:32, 17 June 2020 (UTC)

"Cueball is clearly constructed from lines - which (of course) have no width and therefore zero area and as a consequence, cannot obstruct any photons to cause a shadow to form." Except, his head has to be solid because it completely obscures the lower right corner of the projection frame. 162.158.78.10 01:39, 21 June 2020 (UTC)

It is clearly Randall that makes this presentation based on the caption. Have added this to the explanation and transcript --Kynde (talk) 09:32, 17 June 2020 (UTC)

Proof of the inconsistency of arithmetic

Regarding [1], could a professional number theorist please opine on the proof? And for that matter, is Peano arithmetic inconsistency that bad? If so, is it bad on the scale of 2020? I mean, if there are so many things equivalent to Peano arithmetic, then maybe one of them with a very slight change is consistent? 162.158.107.17 09:55, 17 June 2020 (UTC)

Imagine the meeting! It would be like if aliens were discovered. "Gentlemen, sooner or later it's going to leak that arithmetic is inconsistent. We need plans. Contingency plans! Get to work!" This could make the CDC zombies site look like a test run. We could have the Count muppet with a thirty minute speech capitulating to veganism. 172.68.189.179 21:37, 18 June 2020 (UTC)

In any other year, the inconsistency of arithmetic would cause the collapse of civilization. In 2020, it's keeping it up. 162.158.106.178 21:00, 20 June 2020 (UTC)

[2] 172.68.141.80 03:41, 30 June 2020 (UTC)