{"id":2214,"date":"2018-04-16T11:36:44","date_gmt":"2018-04-16T08:36:44","guid":{"rendered":"https:\/\/news.ubbcluj.ro\/hu\/?p=2214"},"modified":"2018-04-16T11:39:33","modified_gmt":"2018-04-16T08:39:33","slug":"bbte-s-matematikus-kozremukodesevel-szulethetett-attores-a-metrikus-geometria-teruleten","status":"publish","type":"post","link":"https:\/\/news.ubbcluj.ro\/hu\/bbte-s-matematikus-kozremukodesevel-szulethetett-attores-a-metrikus-geometria-teruleten\/","title":{"rendered":"BBTE-s matematikus k\u00f6zrem\u0171k\u00f6d\u00e9s\u00e9vel sz\u00fclethetett \u00e1tt\u00f6r\u00e9s a metrikus geometria ter\u00fclet\u00e9n[*]"},"content":{"rendered":"<p style=\"text-align: justify;\"><strong>A BBTE K\u00f6zgazdas\u00e1g \u00e9s Gazd\u00e1lkod\u00e1studom\u00e1nyi Kara sepsiszentgy\u00f6rgyi kihelyezett tagozat\u00e1nak matematikus professzora, Krist\u00e1ly S\u00e1ndor is k\u00f6zrem\u0171k\u00f6d\u00f6tt annak a tanulm\u00e1nynak az elk\u00e9sz\u00edt\u00e9s\u00e9ben, mely a John Lott, Karl-Theodor Sturm, valamint a Fields-d\u00edjas C\u00e9dric Villani nev\u00e9vel f\u00e9mjelzett, a metrikus geometria ter\u00fclet\u00e9n m\u00e9rf\u00f6ldk\u0151nek sz\u00e1m\u00edt\u00f3 elm\u00e9let \u00e9rv\u00e9nyess\u00e9g\u00e9vel kapcsolatos k<\/strong><strong>\u00e9rd\u00e9seket<\/strong><strong> helyezi \u00faj kontextusba.<\/strong><\/p>\n<p style=\"text-align: justify;\">A vizsg\u00e1lt probl\u00e9ma gy\u00f6kerei a matematikus k\u00f6r\u00f6kben Dido-feladatk\u00e9nt elh\u00edres\u00fclt klasszikus izoperimetrikus fejt\u00f6r\u0151ig ny\u00falnak vissza. Dido nagyb\u00e1tyj\u00e1hoz, Acerb\u00e1szhoz ment feles\u00e9g\u00fcl, akit vagyona miatt meggyilkoltak. Az \u00f6zvegy Ciprusra menek\u00fclt, majd tov\u00e1bb haj\u00f3zott Afrika Szic\u00edli\u00e1hoz k\u00f6zeli partjaira, ahol azzal a k\u00e9r\u00e9ssel kereste fel a vid\u00e9k uralkod\u00f3j\u00e1t, hogy v\u00e1s\u00e1rolna egy akkora f\u00f6lddarabot, amekkora egy marhab\u0151rrel k\u00f6r\u00fclker\u00edthet\u0151. Az uralkod\u00f3 beleegyezett, \u00e9s nagylelk\u0171en meg is aj\u00e1nd\u00e9kozta Did\u00f3t egy j\u00f3kora marhab\u0151rrel. Dido keskeny cs\u00edkokra szabdalta az \u00e1llatb\u0151rt, a cs\u00edkokat \u00f6sszecsom\u00f3zta, \u00edgy olyan hossz\u00fa k\u00f6t\u00e9lhez jutott, amellyel j\u00f3val nagyob f\u00f6ldter\u00fcletet lehetett elker\u00edteni, mint amekkor\u00e1t az uralkod\u00f3 elk\u00e9pzelt. \u00cdgy alap\u00edtotta meg Karth\u00e1g\u00f3 v\u00e1ros\u00e1t. A metrikus geometria nyelv\u00e9n a feladat a k\u00f6vetkez\u0151k\u00e9ppen fogalmazhat\u00f3 meg: adott ker\u00fclet\u0171 s\u00edkidomok k\u00f6z\u00fcl melyiknek legnagyobb a ter\u00fclete? Azaz, keress\u00fck meg azt a legnagyobb ter\u00fcletet, melyet bez\u00e1r egy adott hossz\u00fas\u00e1g\u00fa z\u00e1rt g\u00f6rbe! A Dido-feladat a Brunn\u2013Minkowski-egyenl\u0151tlens\u00e9g k\u00f6vetkezm\u00e9nyek\u00e9nt \u00e9rtelmezhet\u0151, eredetileg ennek az egyenl\u0151tlens\u00e9gnek egy nem eukleideszi igazol\u00e1s\u00e1ra tett k\u00eds\u00e9rlet szolg\u00e1lt a tanulm\u00e1ny kiindul\u00f3pontj\u00e1ul.<\/p>\n<p style=\"text-align: justify;\">Bolyai J\u00e1nos, Nikolai Lobachevsky \u00e9s Carl Fridrich Gauss \u00fatt\u00f6r\u0151 munk\u00e1ja nyom\u00e1n a tizenkilencedik sz\u00e1zadban vil\u00e1goss\u00e1 v\u00e1lt, hogy a g\u00f6rb\u00fclet kiemelt szerepet j\u00e1tszik bizonyos geometriai probl\u00e9m\u00e1k eset\u00e9ben. Bernhard Riemann ugyanebben a sz\u00e1zadban \u00e1llap\u00edtotta meg a g\u00f6rb\u00fclet-tenzor tulajdons\u00e1gait, amelyekre Albert Einsteinnek is elengedhetetlen sz\u00fcks\u00e9ge volt a relativit\u00e1selm\u00e9let kidolgoz\u00e1s\u00e1ban.<\/p>\n<p style=\"text-align: justify;\">Az 1940-es \u00e9vekben Herbert Busemann \u00e9s Aleksandr Danilovich Aleksandrov egym\u00e1st\u00f3l f\u00fcggetlen\u00fcl kezdem\u00e9nyezt\u00e9k a szintetikus geometria, azaz egy olyan m\u00e9rtan kidolgoz\u00e1s\u00e1t, amely a k\u00fcl\u00f6nb\u00f6z\u0151 alakzatok els\u0151sorban projekt\u00edv tulajdons\u00e1gait az analitikus geometria seg\u00e9deszk\u00f6zei (koordin\u00e1t\u00e1k, differenci\u00e1l-strukt\u00far\u00e1k) n\u00e9lk\u00fcl vizsg\u00e1lja. Ennek a paradigm\u00e1nak az egyik le\u00e1gaz\u00e1sa a metrikus geometria, amely a t\u00e9r metrik\u00e1j\u00e1t le\u00edr\u00f3 \u00f6sszef\u00fcgg\u00e9sek igazol\u00e1s\u00e1ra helyezi a hangs\u00falyt.<\/p>\n<p style=\"text-align: justify;\">K\u00f6zel egy \u00e9vtizede John Lott, Karl-Theodor Sturm, valamint C\u00e9dric Villani azzal \u00e9rtek el jelent\u0151s \u00e1tt\u00f6r\u00e9st a szintetikus geometria ter\u00fclet\u00e9n, hogy bevezett\u00e9k az entr\u00f3pia-egyenl\u0151tlens\u00e9gre alapozott g\u00f6rb\u00fcletdimenzi\u00f3-felt\u00e9telt, amelyben az adott geometriai strukt\u00fara m\u00e9ly g\u00f6rb\u00fcleti inform\u00e1ci\u00f3kat takar. A Lott\u2013Sturm\u2013Villani-elm\u00e9let az \u00e1ltalunk tapasztalt vil\u00e1got olyan egyenl\u0151tlens\u00e9gek \u2013 p\u00e9ld\u00e1ul a Brunn\u2013Minkowski, a Borell\u2013Brascamp\u2013Lieb vagy a Bishop\u2013Gromov \u2013 seg\u00edts\u00e9g\u00e9vel jelen\u00edti meg, amelyek szoros kapcsolatban \u00e1llnak az izoperimetrikus egyenl\u0151tlens\u00e9gekkel. H\u00e1tter\u00e9ben olyan Riemann-t\u00e9rmodellek h\u00faz\u00f3dnak, amelyeknek a g\u00f6rb\u00fclete \u00e1lland\u00f3 (ld. g\u00f6mb, eukleid\u00e9szi t\u00e9r vagy hiperbolikus t\u00e9r). 2010-ben Villanit Fields-d\u00edjjal t\u00fcntett\u00e9k ki eredm\u00e9nyei\u00e9rt, az elismer\u00e9s \u2013 az Abel-d\u00edj mellett \u2013 egyfajta matematikai Nobel-d\u00edjnak min\u0151s\u00fcl, amely a negyvenedik \u00e9let\u00e9v\u00fcket m\u00e9g be nem t\u00f6lt\u00f6tt szakembereknek \u00edt\u00e9lhet\u0151 oda.<\/p>\n<p style=\"text-align: justify;\">2009-ben Nicolas Juillet h\u00edvta fel a figyelmet arra, hogy az elm\u00e9let nem m\u0171k\u00f6dik p\u00e9ld\u00e1ul a szubriemanni sokas\u00e1gok legegyszer\u0171bb strukt\u00far\u00e1i, a Heisenberg-csoportok eset\u00e9ben. Ezeket a strukt\u00far\u00e1kat t\u00f6bbek k\u00f6zt a rep\u00fcl\u0151g\u00e9pek p\u00e1ly\u00e1inak vagy a g\u00e9pkocsik mozg\u00e1s\u00e1nak a modellez\u00e9s\u00e9ben szokt\u00e1k haszn\u00e1lni. Juillet c\u00e1folata nyom\u00e1n szakmai k\u00f6r\u00f6kben elfogadott\u00e1 v\u00e1lt az a n\u00e9zet, miszerint, szubriemanni szintr\u0151l szeml\u00e9lve a probl\u00e9m\u00e1t, nincs es\u00e9ly arra, hogy a geometriai alapegyenl\u0151tlens\u00e9geket olyan \u00e1ltal\u00e1nosan t\u00e1rgyaljuk, ahogyan azt a Lott\u2013Sturm\u2013Villani-elm\u00e9let tette.<\/p>\n<p style=\"text-align: justify;\">Balogh Zolt\u00e1nnak, Krist\u00e1ly S\u00e1ndornak \u00e9s Sipos King\u00e1nak t\u00f6bb\u00e9ves kutat\u00e1s nyom\u00e1n siker\u00fclt bizony\u00edtania, hogy a Lott, Sturm, illetve Villani nev\u00e9vel f\u00e9mjelzett elm\u00e9let \u00e1ll\u00edt\u00e1saival anal\u00f3g meg\u00e1llap\u00edt\u00e1sok igazolhat\u00f3k a Heisenberg-csoportok eset\u00e9ben is, ehhez azonban egy, a Villani\u00e9k\u00e9val p\u00e1rhuzamos geometriai kontextusra van sz\u00fcks\u00e9g. Az optim\u00e1lis anyagsz\u00e1ll\u00edt\u00e1s elm\u00e9let\u00e9vel, valamint a Heisenberg-csoportok egzakt riemanni approxim\u00e1ci\u00f3j\u00e1val kimutatt\u00e1k, hogy l\u00e9tezik a probl\u00e9m\u00e1nak egy \u00fagynevezett szubriemanni t\u00e1rgyal\u00e1sa, amellyel teljes m\u00e9rt\u00e9kben le\u00edrhat\u00f3 a szubriemanni vil\u00e1g geometri\u00e1ja.<\/p>\n<p style=\"text-align: justify;\">A <em>Calculus of Variations and Partial Differential Equations<\/em> c\u00edm\u0171 szaklapban k\u00f6z\u00f6lt eredm\u00e9nyt C\u00e9dric Villani is m\u00e9ltatta, el\u0151rebocs\u00e1tva, hogy ez az eredm\u00e9ny lehet a hi\u00e1nyz\u00f3 l\u00e1ncszem a riemanni, a finsleri, valamint a szubriemanni geometriai egyenl\u0151tlens\u00e9gek egys\u00e9ges\u00edt\u00e9s\u00e9ben.<\/p>\n<h5 style=\"text-align: justify;\">*Zolt\u00e1n M. BALOGH, Alexandru KRIST\u00c1LY, Kinga SIPOS (2018): Geometric inequalities on Heisenberg groups. <em>Calculus of Variations and Partial Differential Equations<\/em>, 57\/61.<\/h5>\n<figure id=\"attachment_2215\" aria-describedby=\"caption-attachment-2215\" style=\"width: 596px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/news.ubbcluj.ro\/hu\/wp-content\/uploads\/2018\/04\/Clipboard01.jpg\" data-rel=\"lightbox-gallery-nB7IvxNi\" data-rl_title=\"\" data-rl_caption=\"\" title=\"\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2215\" src=\"https:\/\/news.ubbcluj.ro\/hu\/wp-content\/uploads\/2018\/04\/Clipboard01.jpg\" alt=\"\" width=\"596\" height=\"373\" data-id=\"2215\" \/><\/a><figcaption id=\"caption-attachment-2215\" class=\"wp-caption-text\">A k\u00e9pen a Fields-d\u00edjas C\u00e9dric Villani l\u00e1that\u00f3. Forr\u00e1s: http:\/\/synestheorie.fr\/2014\/11\/26\/musique-mathematiques-france-musique\/<\/figcaption><\/figure>\n<p>A tanulm\u00e1nyismertet\u0151t Serest\u00e9ly Zal\u00e1n szerkesztette.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A BBTE K\u00f6zgazdas\u00e1g \u00e9s Gazd\u00e1lkod\u00e1studom\u00e1nyi Kara sepsiszentgy\u00f6rgyi kihelyezett tagozat\u00e1nak matematikus professzora, Krist\u00e1ly S\u00e1ndor is k\u00f6zrem\u0171k\u00f6d\u00f6tt annak a tanulm\u00e1nynak az elk\u00e9sz\u00edt\u00e9s\u00e9ben, mely a John Lott, Karl-Theodor Sturm, valamint a Fields-d\u00edjas C\u00e9dric Villani nev\u00e9vel f\u00e9mjelzett, a metrikus geometria ter\u00fclet\u00e9n m\u00e9rf\u00f6ldk\u0151nek sz\u00e1m\u00edt\u00f3 elm\u00e9let &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/news.ubbcluj.ro\/hu\/bbte-s-matematikus-kozremukodesevel-szulethetett-attores-a-metrikus-geometria-teruleten\/\"> <span class=\"screen-reader-text\">BBTE-s matematikus k\u00f6zrem\u0171k\u00f6d\u00e9s\u00e9vel sz\u00fclethetett \u00e1tt\u00f6r\u00e9s a metrikus geometria ter\u00fclet\u00e9n[*]<\/span> Tov\u00e1bb &raquo;<\/a><\/p>\n","protected":false},"author":9,"featured_media":2215,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,6,52,12,7],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.9.1 - 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