The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I’ve been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it’s a big deal lately?
\nI’m definitely not the best person to ask; the reason for the recent attention to the Poincarė conjecture is deep topology, which is not one of my stronger fields. But I’ll give it my best shot. (It’s actually rather bad timing. I’m planning on starting to write about topology later this week; and since the Poincarė conjecture is specifically about topology, it really wouldn’t have hurt to have introduced some topology first. But that’s how the cookie crumbles, eh?)
\nSo what is it?
\n—————–
\nIn 1904, the great mathematician Henri Poincarė was studying topology, and came up with an interesting question.
\nWe know that if we look at closed two-dimensional surfaces forming three dimensional shapes (manifolds), that if the three dimensional shape has no holes in it, then it’s possible to transform it by bending, twisting, and stretching – but *without tearing* – into a sphere.
\nPoincarė wondered about higher dimensions. What about a three dimensional closed surface in a four-dimensional space (a 3-manifold)? Or a closed 4-manifold?
\nThe conjecture, expressed *very* loosely and imprecisely, was that in any number of dimensions *n*, any figure without holes could be reduced to an *n*-dimensional sphere.
\nIt’s trivial to show that that’s true for 2-dimensional surfaces in a three dimensional space; that is, that all closed 2-dimensional surfaces without holes can be transformed without tearing into our familiar sphere (which topologists call a 2-sphere, because it’s got a two dimensional surface).
\nFor surfaces with more than two dimensions, it becomes downright mind-bogglingly difficult. And in fact, it turns out to be *hardest* to prove this for the 3-sphere. Nearly every famous mathematician of the 20th century took a stab at it, and all of them failed. (For example, Whitehead of the infamous Russell & Whitehead “Principia” published an incorrect proof in 1934.)
\nWhy is it so hard?
\n——————
\nVisualizing the shapes of closed 2-manifolds is easy. They form familiar figures in three dimensional space. We can imagine grabbing them, twisting them, stretching them. We can easily visualize almost anything that you can do with a closed two-dimensional surface. So reasoning about them is very natural to us.
\nBut what about a “surface” that is itself three dimensional, forming a figure that takes 4 dimensions. What does it look like? What does *stretching* it mean? What is does a hole in a 4-dimensional shape look like? How can I tell if a particular complicated figure is actually just something tied in knots to make it look complicated, or if it actually has holes in it? What are the possible shapes of things in 4, 5, 6 dimensions?
\nThat’s basically the problem. The math of it is generally expressed rather differently, but what it comes down to is that we don’t have a good intuitive sense of what transformations and what shapes really work in more than three dimensions.
\nWhat’s the big deal lately?
\n——————————-
\nThe conjecture was proved for all surfaces with seven or more dimensions in 1960. Five and six dimensions followed only two years later, proven in 1962. It took another 20 years to find a proof for 4 dimensions, which was finally done in 1982. Since 1982, the only open question was the 3- manifold. Was the Poincarė conjecture true for all dimensions?
\nThere’s a million dollar reward for answer to that question with a correct proof; and each of the other proofs of the conjecture for higher dimensions won the mathematical equivalent of the Nobel Prize. So the rewards for figuring out the answer and proving it are enormous.
\nIn 2003, a rather strange reclusive Russian mathematician named Grigory Perelman published a proof of a *stronger* version of the Poincarė conjecture under the incredibly obvious title “The Entropy Formula for the Ricci Flow and Its Geometric Application”.
\nIt’s taken 3 years for people to work through the proof and all of its details in order to verify its correctness. In full detail, it’s over 1000 pages of meticulous mathematical proof, so verifying its correctness is not exactly trivial. But now, three years later, to the best of my knowledge, pretty much everyone is pretty well convinced of its correctness.
\nSo what’s the basic idea of the proof? This is *so* far beyond my capabilities that it’s almost laughable for me to even attempt to explain it, but I’ll give it my best shot.
\nThe Ricci flow is a mathematical transformation which effectively causes a *shrinking* action on a closed metric 3-surface. As it shrinks, it “pinches off” irregularities or kinks in the surface. The basic idea behind the proof is that it shows that the Ricci flow applied to metric 3-surfaces will shrink to a 3-sphere. The open question was about the kinks: will the Ricci flow eliminate all of them? Or are there structures that will *continually* generate kinks, so that the figure never reduces to a 3-sphere?
\nWhat Perelman did was show that all of the possible types of kinks in the Ricci flow of a closed metric 3-surface would eventually all disappear into either a 3-sphere, or a 3-surface with a hole.
\nSo now that we’re convinced of the proof, and people are ready to start handing out the prizes, where’s Professor Perelman?
\n*No one knows*.
\nHe’s a recluse. After the brief burst of fame when he first published his proof, he disappeared into the deep woods in the hinterlands of Russia. The speculation is that he has a cabin back there somewhere, but no one knows. No one knows where to find him, or how to get in touch with him.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Poincarė conjecture has been in the news lately, with an article in the Science Times today. So I’ve been getting lots of mail from people asking me to explain what the Poincarė conjecture is, and why it’s a big deal lately? I’m definitely not the best person to ask; the reason for the recent […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-118","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-1U","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/118","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=118"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/118\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=118"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=118"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}