2320: 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.
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:
- 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).
- The Hodge conjecture in algebraic geometry.
- 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.
- The Riemann hypothesis, which asserts that all non-trivial zeroes of the Riemann zeta function have real part one-half.
- 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.
- 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.
- 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.
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.
- [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.]
- 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.
- Randall misspells Perelman as "Perlman" in the sentence in the panel but spells it correctly in the title-text.
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