{"id":132205,"date":"2024-12-01T18:22:11","date_gmt":"2024-12-01T11:22:11","guid":{"rendered":"https:\/\/hotvideos24.online\/?p=132205"},"modified":"2024-12-01T18:22:11","modified_gmt":"2024-12-01T11:22:11","slug":"maths-bunkbed-conjecture-has-been-proven-false-after-40-years","status":"publish","type":"post","link":"https:\/\/hotvideos24.online\/?p=132205","title":{"rendered":"Math&#8217;s &#8220;Bunkbed Conjecture&#8221; Has Been Proven False After 40 Years"},"content":{"rendered":"<p> <script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-3711241968723425\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-format=\"fluid\"\r\n     data-ad-layout-key=\"-fb+5w+4e-db+86\"\r\n     data-ad-client=\"ca-pub-3711241968723425\"\r\n     data-ad-slot=\"7910942971\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script><br \/>\n<\/p>\n<div>\n<p id=\"isPasted\">For close to 40 years, a simple little hypothesis has been quietly sitting in a corner of graph theory, minding its own business. Known as the \u201cbunkbed conjecture\u201d, it always seemed kind of self-evidently true \u2013 sure, nobody could <em>prove<\/em> it, but it made sense \u2013 and certainly, nobody had ever found a counterexample.\u00a0<\/p>\n<p>Until now. In a surprise to pretty much everybody, a group of mathematicians announced a paper last month that they claim proves the conjecture is false. Currently published on the arXiv preprint server, and thus not yet peer-reviewed, the paper is nevertheless already making waves in the mathematical world \u2013 not only for the proof itself, but for what it says about math as an entire discipline.<\/p>\n<h2><strong>The conjecture<\/strong><\/h2>\n<p>First posited by the physicist Pieter Kasteleyn <a href=\"https:\/\/www.jstor.org\/stable\/2652916\" target=\"_blank\" rel=\"noopener\">to a colleague in 1985<\/a>, the bunkbed conjecture really has nothing to do with beds at all. Rather, it concerns graphs \u2013 and unless you\u2019re a working mathematician, probably not the kind of graphs you\u2019re thinking of right now.<\/p>\n<p>\u201cA graph consists of a bunch of vertices and a bunch of edges that connect the vertices,\u201d explains Trefor Bazett, an Assistant Teaching Professor in the University of Victoria\u2019s Mathematics and Statistics department, in a <a href=\"https:\/\/www.youtube.com\/watch?v=aDU-oTiHnCs\" target=\"_blank\" rel=\"noopener\">recent YouTube video<\/a> about the proof.\u00a0<\/p>\n<div class=\"fr-image-container no-background\" data-asset-id=\"80550\" data-reactroot=\"\">\n<div class=\"fr-image\"><picture title=\"\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-l.webp\" type=\"image\/webp\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-l.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-m.webp\" type=\"image\/webp\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-m.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-s.webp\" type=\"image\/webp\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/a-graph-with-six-vertices-and-eight-edges-s.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><img decoding=\"async\" class=\"inline-image fr-fic fr-dib\" data-asset-id=\"80550\" src=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80550\/graph1-ezgif.com-webp-to-jpg-converter.jpg\" alt=\"A graph with six vertices and eight edges\" title=\"A graph with six vertices and eight edges\" loading=\"lazy\"\/><\/source><\/source><\/source><\/source><\/source><\/source><\/picture><\/div><figcaption class=\"fr-figcaption\">\n<p>A graph with six vertices and eight edges.<\/p>\n<p>Image credit: IFLScience<\/p>\n<\/figcaption><\/div>\n<p>\u201cYou can imagine, perhaps, <a href=\"https:\/\/www.iflscience.com\/rare-breakthrough-in-notoriously-hard-math-problem-means-your-parties-just-got-more-efficient-68271\" target=\"_blank\" rel=\"noopener\">people in a social network<\/a>,\u201d he suggests, \u201cand then the connection is whether or not you\u2019re friends.\u201d<\/p>\n<p>Double this graph exactly, and you can create what\u2019s known as a bunkbed graph: two identical graphs on top of each other, connected by \u201cposts\u201d. Look, once you see it in person, you\u2019ll understand all the themed terminology.<\/p>\n<div class=\"fr-image-container no-background\" data-asset-id=\"80551\" data-reactroot=\"\">\n<div class=\"fr-image\"><picture title=\"\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-l.webp\" type=\"image\/webp\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-l.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-m.webp\" type=\"image\/webp\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-m.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-s.webp\" type=\"image\/webp\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/bunkbed-conjecture-diagram-s.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><img decoding=\"async\" class=\"inline-image fr-fic fr-dib\" data-asset-id=\"80551\" src=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80551\/graph2-ezgif.com-webp-to-jpg-converter.jpg\" alt=\"Bunkbed conjecture diagram\" title=\"Bunkbed conjecture diagram\" loading=\"lazy\"\/><\/source><\/source><\/source><\/source><\/source><\/source><\/picture><\/div><figcaption class=\"fr-figcaption\">\n<p>The same graph put in bunkbed formation. Those pink vertices are called the bedposts.<\/p>\n<p>Image credit: IFLScience<\/p>\n<\/figcaption><\/div>\n<p>So, we have our setup \u2013 our friendships between people, or locations on a map connected by streets, or whatever you\u2019re imagining your graph to represent. Now we\u2019re just going to think about how to move through the graph itself \u2013 so, say you want to get from point <em>u<\/em> to point <em>v<\/em> in our graph above, we have the following options:<\/p>\n<div class=\"fr-image-container no-background\" data-asset-id=\"80552\" data-reactroot=\"\">\n<div class=\"fr-image\"><picture title=\"\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-l.webp\" type=\"image\/webp\"><source media=\"(min-width: 1000px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-l.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-m.webp\" type=\"image\/webp\"><source media=\"(min-width: 568px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-m.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-s.webp\" type=\"image\/webp\"><source media=\"(max-width: 567px)\" srcset=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/bunkbed-conjecture-diagram-s.jpg\" type=\"https:\/\/www.iflscience.com\/image\/jpeg\"><img decoding=\"async\" class=\"inline-image fr-fic fr-dib\" data-asset-id=\"80552\" src=\"https:\/\/assets.iflscience.com\/assets\/articleNo\/76998\/iImg\/80552\/graph3-ezgif.com-webp-to-jpg-converter.jpg\" alt=\"Bunkbed conjecture diagram\" title=\"Bunkbed conjecture diagram\" loading=\"lazy\"\/><\/source><\/source><\/source><\/source><\/source><\/source><\/picture><\/div><figcaption class=\"fr-figcaption\">\n<p>Four potential routes (not exhaustive!) from <em>u<\/em> to <em>v<\/em> highlighted in red.<\/p>\n<p>Image credit: IFLScience<\/p>\n<\/figcaption><\/div>\n<p>With us so far? Great, because this is where things get a bit more complicated. What we\u2019re going to do now is delete some of the edges \u2013 lose friends; block up streets, whatever you like \u2013 and see how likely it is that we can still get from <em>u<\/em> to <em>v<\/em> afterward.<\/p>\n<p>So, with all that background, we can now get to the statement of the bunkbed conjecture, which is this: <em>P<\/em>(<em>u\u00a0<\/em>\u2194 <em>v<\/em>) \u2265<em>\u00a0P<\/em>(<em>u\u00a0<\/em>\u2194 <em>v\u2019<\/em>).<\/p>\n<p>\u201cIt says that the probability that I can get from <em>u<\/em> to <em>v<\/em> \u2013 that is, the probability that I can move along the base \u2013 is bigger or equal to the probability that I can start in the base and then get to <em>v\u2019<\/em> [\u2026] on the top bunk,\u201d Bazett explained.\u00a0<\/p>\n<p>\u201cThe conjecture says that this is true for all connected graphs, and all subsets of bedposts, and all pairs <em>u<\/em> and <em>v<\/em>.\u201d<\/p>\n<p>Intuitively, it makes sense: surely, it\u2019s going to be easier to reach an endpoint on the same level as your starting point than one that requires you to travel up a bedpost as well. Trying a few examples will only bolster that conviction \u2013 unless, that is, you\u2019re willing to construct a graph with a few thousand vertices and edges.<\/p>\n<h2><strong>The proof<\/strong><\/h2>\n<p>Often, in math, disproving a hypothesis is easier than proving it. After all, to <em>prove<\/em> something, you have to show it\u2019s true for every possible example, in all situations \u2013 to <em>disprove<\/em> it, you only have to find a single counterexample.<\/p>\n<p>The problem with the bunkbed conjecture was \u2013 well, nobody was looking for that counterexample. \u201cWhy look for a counterexample if the conjecture is so obviously true?\u201d wrote Igor Pak, a math professor at UCLA and one of the authors of the new paper, in a <a href=\"https:\/\/igorpak.wordpress.com\/2024\/10\/01\/the-bunkbed-conjecture-is-false\/\" target=\"_blank\" rel=\"noopener\">blog post<\/a> on the breakthrough.<\/p>\n<p>\u201cWell, because you always should,\u201d he countered. \u201cFor any conjecture. Especially if everyone else is so <em>sure<\/em>, as in <em>completely absolutely sure without a doubt<\/em>, that the conjecture is true.\u201d<\/p>\n<p>Now, it&#8217;s the 2020s; you know <a href=\"https:\/\/www.iflscience.com\/todays-top-ai-went-up-against-expert-mathematicians-it-lost-badly-76919\" target=\"_blank\" rel=\"noopener\">how math is done these days<\/a>, and so did Pak. \u201cWe started with a myriad of computer experiments trying all small graphs,\u201d he wrote. \u201cWhen those failed, we tried to use AI and other computer assisted tools.\u201d<\/p>\n<p>Even still, no counterexamples seemed to be forthcoming \u2013 and the team started worrying that, even if one did turn up, it wouldn\u2019t be enough to fully disprove the conjecture. The graphs being sampled by the neural networks were so large at that point that calculating the relevant probabilities <em>exactly<\/em> would be impossible, and so any proof would be at best something like 99.9999 percent certain to be correct.<\/p>\n<p>But while \u201c99.99 percent confidence [\u2026] may be a gold standard in nuclear physics,\u201d Pak wrote, \u201cmath journals tend to prefer 100 percent correctness.\u201d<\/p>\n<p>\u201cMost journals would refuse to even <em>consider<\/em> a \u2018<a href=\"https:\/\/home.cern\/resources\/faqs\/five-sigma\" target=\"_blank\" rel=\"noopener\">five sigma<\/a> counterexample\u2019,\u201d he added.<\/p>\n<p>So, rather than persevere with machine learning techniques that weren\u2019t bearing fruit \u2013 and whose results may not be accepted even if it was successful \u2013 the team took a step back. And then, in June of this year, a <a href=\"https:\/\/arxiv.org\/abs\/2406.01790\" target=\"_blank\" rel=\"noopener\">paper<\/a> hit the arXiv which changed everything.<\/p>\n<p>\u201cI found it in the evening, and I read it until 3 a.m.,\u201d Nikita Gladkov, one of Pak\u2019s graduate students and a co-author of the paper, told <a href=\"https:\/\/www.quantamagazine.org\/maths-bunkbed-conjecture-has-been-debunked-20241101\/\" target=\"_blank\" rel=\"noopener\">Quanta<\/a>. \u201cI was like, \u2018Wow, this is crazy. Absolutely mind-boggling.\u2019\u201d<\/p>\n<p>It wasn\u2019t a proof of the bunkbed conjecture exactly, but it was close \u2013 a formulation of the statement which dealt with objects called hypergraphs, rather than graphs. The author, a postgraduate student at the University of Cambridge and <a href=\"https:\/\/googology.fandom.com\/wiki\/Lawrence_Hollom\" target=\"_blank\" rel=\"noopener\">accomplished googologist<\/a> named Lawrence Hollom, had shown that in these objects, the bunkbed conjecture was\u2026 false.<\/p>\n<p>Hollom had presented his work as an attempt to generalize the bunkbed conjecture \u2013 or, as it turned out, to show that it couldn\u2019t be generalized. In the end, though, it was his paper that would inspire the proof of the original conjecture.<\/p>\n<p>By converting Hollom\u2019s hypergraph, the team created a graph that could potentially disprove the bunkbed conjecture. It was absolutely monstrous \u2013 7,222 vertices, connected by 14,442 edges \u2013 and the difference in the relevant probabilities was minute: \u201castronomically small,\u201d Pak wrote, \u201con the order of -10<sup>-6500<\/sup>.\u201d\u00a0<\/p>\n<p>\u201cBut it\u2019s negative, which is all we need,\u201d he added. The conjecture was officially false.<\/p>\n<h2><strong>The upshot<\/strong><\/h2>\n<p>So, what does this mean, other than the obvious? Well, there are some disappointments, particularly for applied mathematicians and physicists: had the bunkbed conjecture turned out to be true, it would have validated a widely believed assumption about how fluids travel through solids, and given a handhold to researchers investigating the physics of percolation.<\/p>\n<p>But more than that, there are the moral impacts of the breakthrough. Should future mathematicians be more willing to accept probabilistic proofs? Would they be as valid, or as complete?<\/p>\n<p>\u201cIt\u2019s a philosophical question,\u201d Noga Alon, a math professor at Princeton, told Quanta. \u201cHow do we view proofs that are only true with high probability?\u201d\u00a0<\/p>\n<p>\u201cMaybe a probabilistic proof would give you less understanding or intuition of what\u2019s really going on,\u201d he said.<\/p>\n<p>Finally, it\u2019s a warning to mathematicians not to accept a conjecture just because they like it. \u201cWe have to be suspicious, even about things that intuitively look very likely to be true,\u201d Alon said.<\/p>\n<p>It\u2019s a sentiment that Pak has long championed. \u201cSome conjectures are motivated by substance,\u201d he told Quanta, \u201cand other conjectures are motivated by wishful thinking.\u201d<\/p>\n<p>The bunkbed conjecture, it seems, was the latter.<\/p>\n<p>The paper, which is not yet peer-reviewed, can be found on the <a href=\"https:\/\/arxiv.org\/abs\/2410.02545\" target=\"_blank\" rel=\"noopener\">arXiv<\/a>.<\/p>\n<\/div>\n<p><script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-3711241968723425\"\r\n     crossorigin=\"anonymous\"><\/script>\r\n<ins class=\"adsbygoogle\"\r\n     style=\"display:block\"\r\n     data-ad-format=\"fluid\"\r\n     data-ad-layout-key=\"-fb+5w+4e-db+86\"\r\n     data-ad-client=\"ca-pub-3711241968723425\"\r\n     data-ad-slot=\"7910942971\"><\/ins>\r\n<script>\r\n     (adsbygoogle = window.adsbygoogle || []).push({});\r\n<\/script><br \/>\n<br \/><div data-type=\"_mgwidget\" data-widget-id=\"1660802\">\r\n<\/div>\r\n<script>(function(w,q){w[q]=w[q]||[];w[q].push([\"_mgc.load\"])})(window,\"_mgq\");\r\n<\/script>\r\n<br \/>\n<br \/><a href=\"https:\/\/www.iflscience.com\/maths-bunkbed-conjecture-has-been-proven-false-after-40-years-76998\">Source link <\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>For close to 40 years, a simple little hypothesis has been quietly sitting in a corner of graph theory, minding its own business. Known as the \u201cbunkbed conjecture\u201d, it always &hellip; <a href=\"https:\/\/hotvideos24.online\/?p=132205\" class=\"more-link\">Read More<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8628],"tags":[],"class_list":["post-132205","post","type-post","status-publish","format-standard","hentry","category-science","entry"],"_links":{"self":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts\/132205","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=132205"}],"version-history":[{"count":0,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=\/wp\/v2\/posts\/132205\/revisions"}],"wp:attachment":[{"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=132205"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=132205"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hotvideos24.online\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=132205"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}