{"id":17918,"date":"2023-10-06T12:25:51","date_gmt":"2023-10-06T04:25:51","guid":{"rendered":"https:\/\/virtual.math.ntnu.edu.tw\/?p=17918"},"modified":"2023-10-06T12:25:51","modified_gmt":"2023-10-06T04:25:51","slug":"lecture_20231011","status":"publish","type":"post","link":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/2023\/10\/06\/lecture_20231011\/","title":{"rendered":"[\u5c08\u984c\u6f14\u8b1b] <\/span>\u301010\u670811\u65e5\u3011Shagnik Das \/ Covering grids with multiplicity"},"content":{"rendered":"\t\t
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Covering grids with multiplicity<\/h3>\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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\u6642\u3000\u9593\uff1a2023-10-11 14:00 (\u661f\u671f\u4e09) \/ \u5730\u3000\u9ede\uff1aS101 \/ \u8336\u3000\u6703\uff1aS205 (13:30)<\/span><\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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Shagnik Das
\u6234\u5c1a\u5e74 \u6559\u6388<\/div>
(National Taiwan University)<\/div>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t
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It is a fairly simple exercise to determine the minimum number of hyperplanes needed to cover all the points of a finite grid S<\/mi>1<\/mn><\/msub><\/mrow>S_1<\/annotation><\/semantics><\/math><\/span>S<\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> x S<\/mi>2<\/mn><\/msub><\/mrow>S_2<\/annotation><\/semantics><\/math><\/span>S<\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> x … x S<\/mi>d<\/mi><\/msub>\u2208<\/mo>F<\/mi>d<\/mi><\/msup><\/mrow>S_d \\in \\mathbb{F}^d<\/annotation><\/semantics><\/math><\/span>S<\/span>d<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span>F<\/span>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>. However, the situation becomes much more complex if you add the condition that one of the points of the grid must be avoided, leading to a classic problem in combinatorial geometry. This was resolved by Alon and F\u00fcredi in a classic result that popularised the use of algebraic methods in combinatorics.<\/p>

Recent work of Clifton and Huang, in which they considered the question a variant of the problem where the nonzero points of a hypercube should be covered multiple times while avoiding the origin, brought renewed interest to this problem. In addition to Alon-F\u00fcredi-style algebraic arguments, they used linear programming to asymptotically resolve the problem in certain ranges.<\/p>

Despite all the attention this problem has received, there remain many open problems, even in the case of two-dimensional grids over R<\/mi><\/mrow>\\mathbb{R}<\/annotation><\/semantics><\/math><\/span>R<\/span><\/span><\/span><\/span><\/span>. In this talk, after surveying the background and introducing the techniques used, we shall present some recent results that resolve the problem for almost all two-dimensional grids.<\/p>

The new results are joint work with Anurag Bishnoi, Simona Boyadzhiyska and Yvonne den Bakker, and separately with Valjakas Djaljapayan, Yen-Chi Roger Lin and Wei-Hsuan Yu.<\/p>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t

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\u53cb\u5584\u5217\u5370<\/a><\/p>","protected":false},"excerpt":{"rendered":"

Covering grids with multiplicity \u6642\u3000\u9593\uff1a2023-10-11 14:00 ( […]<\/p>\n","protected":false},"author":23,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1,124],"tags":[],"_links":{"self":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/17918"}],"collection":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/users\/23"}],"replies":[{"embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/comments?post=17918"}],"version-history":[{"count":4,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/17918\/revisions"}],"predecessor-version":[{"id":17925,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/posts\/17918\/revisions\/17925"}],"wp:attachment":[{"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/media?parent=17918"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/categories?post=17918"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/virtual.math.ntnu.edu.tw\/index.php\/wp-json\/wp\/v2\/tags?post=17918"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}