{"629562":{"#nid":"629562","#data":{"type":"news","title":"EXTERNAL NEWS - Professor Makes Breakthrough in Cap Set Problem","body":[{"value":"\u003Cp\u003E\u003Ca href=\u0022http:\/\/people.math.gatech.edu\/~ecroot\/\u0022 target=\u0022_blank\u0022\u003EErnie Croot\u003C\/a\u003E\u0026nbsp;of the Georgia Institute of Technology,\u0026nbsp;\u003Ca href=\u0022http:\/\/math.haifa.ac.il\/seva\/\u0022 target=\u0022_blank\u0022\u003EVsevolod Lev\u003C\/a\u003E\u0026nbsp;of the University of Haifa, Oranim, in Israel, and\u0026nbsp;\u003Ca href=\u0022https:\/\/www.cs.bme.hu\/~ppp\/indexen.html\u0022 target=\u0022_blank\u0022\u003EP\u0026eacute;ter P\u0026aacute;l Pach\u003C\/a\u003E\u0026nbsp;of the Budapest University of Technology and Economics in Hungary, have made a major breakthrough in the \u003Ca href=\u0022https:\/\/en.wikipedia.org\/wiki\/Cap_set\u0022\u003Ecap set problem\u003C\/a\u003E.\u003C\/p\u003E\r\n\r\n\u003Cp\u003ERead the full research paper here:\u003Cbr \/\u003E\r\n\u003Cbr \/\u003E\r\n\u003Ca href=\u0022https:\/\/arxiv.org\/abs\/1605.01506\u0022\u003Ehttps:\/\/arxiv.org\/abs\/1605.01506\u003C\/a\u003E\u003Cbr \/\u003E\r\n\u003Cbr \/\u003E\r\nRead the full\u0026nbsp;Quanta article here:\u003Cbr \/\u003E\r\n\u003Cbr \/\u003E\r\n\u003Ca href=\u0022https:\/\/www.quantamagazine.org\/set-proof-stuns-mathematicians-20160531\/\u0022\u003Ehttps:\/\/www.quantamagazine.org\/set-proof-stuns-mathematicians-20160531\/\u003C\/a\u003E\u003C\/p\u003E\r\n\r\n\u003Cp\u003EHighlights from the Quanta article:\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThe result is related to the card game SET, in which cards with four attributes - color (red, purple, or green), shape (oval, diamond, or squiggle), shading (solid, striped, or empty), and count (one, two, or three) - need to be matched with 3 cards being a match, or \u0026quot;set\u0026quot;, if all three cards either all share or all differ in all three attributes.\u003C\/p\u003E\r\n\r\n\u003Cblockquote\u003E\r\n\u003Cp\u003EIn a series of papers posted online in recent weeks, mathematicians have solved a problem about the pattern-matching card game Set that predates the game itself. The solution, whose simplicity has stunned mathematicians, is already leading to advances in other\u0026nbsp;\u003Ca href=\u0022https:\/\/www.quantamagazine.org\/tag\/combinatorics\/\u0022\u003Ecombinatorics\u003C\/a\u003E\u0026nbsp;problems.\u003C\/p\u003E\r\n\u003C\/blockquote\u003E\r\n\r\n\u003Cp\u003EThe result extends previous results in an unexpected leap, while the\u0026nbsp;proof is quite profound in its simplicity, and is likely to have far reaching implications.\u003C\/p\u003E\r\n\r\n\u003Cblockquote\u003E\r\n\u003Cp\u003EThe proof uses the \u0026ldquo;polynomial method,\u0026rdquo; an innovation that, despite its simplicity, only rose to prominence on the mathematical scene about a decade ago. The approach produces \u0026ldquo;beautiful short proofs,\u0026rdquo; Tao said. It\u0026rsquo;s \u0026ldquo;sort of magical.\u0026rdquo;\u003C\/p\u003E\r\n\u003C\/blockquote\u003E\r\n\r\n\u003Cblockquote\u003E\r\n\u003Cp\u003EThe previous results were \u0026ldquo;already considered to be quite a big breakthrough, but this completely smashes the bounds that they achieved,\u0026rdquo; said\u0026nbsp;\u003Ca href=\u0022https:\/\/www.dpmms.cam.ac.uk\/~wtg10\/\u0022 target=\u0022_blank\u0022\u003ETimothy Gowers\u003C\/a\u003E, a mathematician and Fields medalist at the University of Cambridge.\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThere\u0026rsquo;s still room to improve the bound on cap sets, but in the near term, at least, any further progress is likely to be incremental, Gowers said. \u0026ldquo;In a certain sense this completely finishes the problem.\u0026rdquo;\u003C\/p\u003E\r\n\u003C\/blockquote\u003E\r\n\r\n\u003Cblockquote\u003E\r\n\u003Cp\u003EThe paper soon set off a cascade of what Ellenberg called \u0026ldquo;math at Internet speed.\u0026rdquo;\u0026nbsp; Within 10 days, Ellenberg and\u0026nbsp;\u003Ca href=\u0022http:\/\/homepage.tudelft.nl\/64a8q\/\u0022 target=\u0022_blank\u0022\u003EDion Gijswijt\u003C\/a\u003E, a mathematician at Delft University of Technology in the Netherlands, had each independently\u0026nbsp;\u003Ca href=\u0022https:\/\/quomodocumque.files.wordpress.com\/2016\/05\/cap-set.pdf\u0022 target=\u0022_blank\u0022\u003Eposted papers\u003C\/a\u003E\u0026nbsp;\u003Ca href=\u0022http:\/\/homepage.tudelft.nl\/64a8q\/progressions.pdf\u0022 target=\u0022_blank\u0022\u003Eshowing how\u003C\/a\u003E\u0026nbsp;to modify the argument to polish off the original cap set problem in just three pages.\u003C\/p\u003E\r\n\u003C\/blockquote\u003E\r\n\r\n\u003Cblockquote\u003E\r\n\u003Cp\u003EMathematicians are now scrambling to figure out the implications of the new proof. Already,\u0026nbsp;\u003Ca href=\u0022http:\/\/arxiv.org\/abs\/1605.06702\u0022 target=\u0022_blank\u0022\u003Ea paper has been posted online\u003C\/a\u003E\u0026nbsp;showing that the proof rules out one of the approaches mathematicians were using to try to create more efficient matrix multiplication algorithms. And on May 17,\u0026nbsp;\u003Ca href=\u0022http:\/\/www.ma.huji.ac.il\/~kalai\/\u0022 target=\u0022_blank\u0022\u003EGil Kalai\u003C\/a\u003E, of the Hebrew University of Jerusalem, wrote an\u0026nbsp;\u003Ca href=\u0022https:\/\/gilkalai.wordpress.com\/2016\/05\/17\/polymath-10-emergency-post-5-the-erdos-szemeredi-sunflower-conjecture-is-now-proven\/\u0022 target=\u0022_blank\u0022\u003E\u0026ldquo;emergency\u0026rdquo; blog post\u003C\/a\u003E\u0026nbsp;pointing out that the cap set result can be used to prove the \u0026ldquo;Erd\u0151s-Szemer\u0026eacute;di sunflower conjecture,\u0026rdquo; which concerns sets that overlap in a sunflower pattern.\u003C\/p\u003E\r\n\r\n\u003Cp\u003E\u0026ldquo;I think a lot of people will be thinking, \u0026lsquo;What can I do with this?\u0026rsquo;\u0026rdquo; Gowers said. Croot, Lev and Pach\u0026rsquo;s approach, he wrote in a\u0026nbsp;\u003Ca href=\u0022https:\/\/gowers.wordpress.com\/2016\/05\/19\/reflections-on-the-recent-solution-of-the-cap-set-problem-i\/\u0022 target=\u0022_blank\u0022\u003Eblog post\u003C\/a\u003E, is \u0026ldquo;a major new technique to add to the toolbox.\u0026rdquo;\u003C\/p\u003E\r\n\r\n\u003Cp\u003EThe fact that the cap set problem finally yielded to such a simple technique is humbling, Ellenberg said. \u0026ldquo;It makes you wonder what else is actually easy.\u0026rdquo;\u003C\/p\u003E\r\n\u003C\/blockquote\u003E\r\n\r\n\u003Cp\u003E\u0026nbsp;\u003C\/p\u003E\r\n","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":"","field_summary_sentence":[{"value":"SoM Professor Ernie Croot and others have make a major breakthrough in the cap set problem with a simple proof that is likely to have far reaching implications. 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