{"id":1457,"date":"2015-11-19T04:58:44","date_gmt":"2015-11-19T04:58:44","guid":{"rendered":"https:\/\/www.science.nus.edu.sg\/?p=1457"},"modified":"2019-11-07T05:00:06","modified_gmt":"2019-11-07T05:00:06","slug":"space-symmetries-dictate-the-geometry","status":"publish","type":"post","link":"https:\/\/www.science.nus.edu.sg\/blog\/2015\/11\/space-symmetries-dictate-the-geometry\/","title":{"rendered":"Space Symmetries Dictate the Geometry"},"content":{"rendered":"<p><span><span class=\"style3\">19 Nov 2015 NUS mathematicians discover the relation between space symmetries and the underlying geometric shape of the space<\/span><span class=\"style3\">.<\/span><\/span><\/p>\n<p><span>It is known that a complex torus (or doughnut shaped space) has many space symmetries. Is the converse true? <\/span><\/p>\n<p><span>A team led by Prof De-Qi ZHANG from the Department of Mathematics in NUS has conjectured that a compact complex space of dimension n at least 3 could have a maximal number of dynamically interesting commutative symmetries (i.e., n-1 of them) only when the space is generically identical (i.e. birationally isomorphic) to a complex torus or its quotient space. In the two papers in the references below, his team shows that the above mentioned conjecture holds true unless the space is rationally connected, i.e., covered by 2-dimensional spheres.<\/span><\/p>\n<p><span>This is a breakthrough in understanding the dynamically interesting symmetries of compact spaces. For instance, the above mentioned results of the papers convincingly explain the reason why interesting examples of symmetries discovered so far by many mathematicians in the world are mostly constructed from complex tori by surgeries of blowups or blowdowns or by taking their quotient spaces. It also guides people toward the right spaces (i.e., tori and rationally connected spaces) to find varieties with dynamically interesting symmetries. <\/span><\/p>\n<p><span>Of course, one still has to understand the symmetries on rationally connected spaces, and to find the criteria or obstructions for these special spaces to be generically complex tori or their quotients. The same team is currently working on this.<\/span><\/p>\n<p><span>\u00a0<img fetchpriority=\"high\" decoding=\"async\" alt=\"ZhangDQ Nov\" src=\"https:\/\/www.science.nus.edu.sg\/wp-content\/uploads\/2019\/11\/ZhangDQ_Nov.jpg\" width=\"292\" height=\"235\" \/><\/span><\/p>\n<p><span>Figure shows the torus has horizontal and also vertical rotations as symmetries. [Image credit: <a href=\"http:\/\/www.horntorus.com\/illustration\/standard_dynamic_horn-torus_still.html\">http:\/\/www.horntorus.com\/illustration\/standard_dynamic_horn-torus_still.html<\/a>]<\/span><\/p>\n<p><span><strong>References<\/strong><\/span><\/p>\n<p><span>1. Zhang DQ. \u201cn-dimensional projective varieties with the action of an abelian group of rank n-1.\u201d Transactions of the American Mathematical Society, <a href=\"http:\/\/arxiv.org\/abs\/1412.5779\" title=\"Abstract\">arXiv:1412.5779<\/a>.<span class=\"apple-converted-space\"><\/span><\/span><\/p>\n<p class=\"Default\"><span>2. F Hu, TC Dinh, DQ Zhang. \u201cCompact K\u00e4hler manifolds admitting large solvable groups of automorphisms.\u201d Advances in Mathematics 281 (2015) 33.<\/span><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>19 Nov 2015 NUS mathematicians discover the relation between space symmetries and the underlying geometric shape of the space. It&#8230;<\/p>\n","protected":false},"author":1,"featured_media":1458,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[82,13],"tags":[],"class_list":["post-1457","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-pure-mathematics","category-research-news"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v23.6 (Yoast SEO v23.6) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Space Symmetries Dictate the Geometry - NUS Faculty of Science<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.science.nus.edu.sg\/blog\/2015\/11\/space-symmetries-dictate-the-geometry\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Space Symmetries Dictate the Geometry\" \/>\n<meta property=\"og:description\" content=\"19 Nov 2015 NUS mathematicians discover the relation between space symmetries and the underlying geometric shape of the space. 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