{"id":2253,"date":"2021-12-30T12:17:56","date_gmt":"2021-12-30T10:17:56","guid":{"rendered":"https:\/\/mathematics.haifa.ac.il\/?p=2253"},"modified":"2022-01-06T18:19:03","modified_gmt":"2022-01-06T16:19:03","slug":"u-haifa-topology-geometry-seminar-sunday-jan-2-2022-speaker-dan-mangoubi-hebrew-university-title-a-local-version-of-courants-nodal-domain-theorem","status":"publish","type":"post","link":"https:\/\/mathematics.haifa.ac.il\/?p=2253","title":{"rendered":"U. Haifa Topology &#038; Geometry seminar: Sunday, Jan. 2, 2022. Speaker: Dan Mangoubi (Hebrew University). &#8220;Title: A Local version of Courant&#8217;s Nodal domain Theorem&#8221;"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Time: <\/strong>Sunday Jan 2nd, 17:00 (Israel Time)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Location: <\/strong>On Zoom<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Speaker: <\/strong>Dan Mangoubi (Hebrew University)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Title:<\/strong> A Local version of Courant&#8217;s Nodal domain Theorem.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Slides:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/mathematics.haifa.ac.il\/wp-content\/uploads\/Haifa-University-020122-slides.pdf\">https:\/\/mathematics.haifa.ac.il\/wp-content\/uploads\/Haifa-University-020122-slides.pdf<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Abstract.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding to the k-th eigenvalue.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is easy to see (by direct calculation) that the number of zeros of u_k is exactly k+1.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Equivalently, the number of connected components of the complement of $u_k=0$ is $k$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In 1923 Courant found that in higher dimensions (considering eigenfunctions of the Laplacian on a closed Riemannian manifold M)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">the number of connected components of the open set $M\\setminus {u_k=0}$ is at most $k$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In 1988 Donnelly and Fefferman gave an upper bound on the number of connected components of $B\\setminus {u_k=0}$, where $B$ is a ball in $M$.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">However, their estimate was not sharp (even for spherical harmonics).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We describe the ideas which give the sharp bound on the number of connected components in a ball.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The talk is based on a joint work with S. Chanillo, A. Logunov and E. Malinnikova, with a contribution due to F. Nazarov.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>ZOOM COORDINATES:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/us02web.zoom.us\/j\/84987591026?pwd=aTJWcFlOSldKck5leGt2d25CYkRqZz09\">https:\/\/us02web.zoom.us\/j\/84987591026?pwd=aTJWcFlOSldKck5leGt2d25CYkRqZz09<\/a><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Meeting ID: 849 8759 1026<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Time: Sunday Jan 2nd, 17:00 (Israel Time) Location: On Zoom Speaker: Dan Mangoubi (Hebrew University) Title: A Local version of Courant&#8217;s Nodal domain Theorem. Slides: https:\/\/mathematics.haifa.ac.il\/wp-content\/uploads\/Haifa-University-020122-slides.pdf Abstract. Let u_k be an eigenfunction of a vibrating string (with fixed ends) corresponding&#8230;<br \/><a class=\"read-more-button\" href=\"https:\/\/mathematics.haifa.ac.il\/?p=2253\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,4],"tags":[],"class_list":["post-2253","post","type-post","status-publish","format-standard","hentry","category-gt-seminar","category-seminar"],"_links":{"self":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2253","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2253"}],"version-history":[{"count":2,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2253\/revisions"}],"predecessor-version":[{"id":2268,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=\/wp\/v2\/posts\/2253\/revisions\/2268"}],"wp:attachment":[{"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2253"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2253"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathematics.haifa.ac.il\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2253"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}