{"id":392,"date":"2021-10-04T01:24:29","date_gmt":"2021-10-04T08:24:29","guid":{"rendered":"\/wp\/hades\/?p=392"},"modified":"2021-10-04T01:24:29","modified_gmt":"2021-10-04T08:24:29","slug":"a-stationary-set-method-for-estimating-oscillatory-integrals","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/hades\/2021\/10\/04\/a-stationary-set-method-for-estimating-oscillatory-integrals\/","title":{"rendered":"A stationary set method for estimating oscillatory integrals"},"content":{"rendered":"\n\t\t\t\t\n<p>The HADES seminar on Tuesday,&nbsp;<strong>October 5th<\/strong>,&nbsp;will be given by&nbsp;<strong>Ruixiang Zhang<\/strong> at&nbsp;<strong>5 pm<\/strong>&nbsp;in&nbsp;<strong>740 Evans<\/strong>.<\/p>\n\n\n\n<p><strong>Speaker<\/strong>: Ruixiang Zhang (UC Berkeley)<\/p>\n\n\n\n<p><strong>Abstract:<\/strong> Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \\int_{[0,1]^d} e(P(\\xi)) \\mathrm{d}\\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, I will introduce a &#8220;stationary set&#8221; method that gives an upper bound with simple geometric meaning. The proof of this bound mainly relies on the theory of o-minimal structures. As an application of our bound, we obtain the sharp convergence exponent in the two dimensional Tarry&#8217;s problem for every degree via additional analysis on stationary sets. Consequently, we also prove the sharp $L^{\\infty} \\to L^p$ Fourier extension estimates for every two dimensional Parsell-Vinogradov surface whenever the endpoint of the exponent $p$ is even. This is joint work with Saugata Basu, Shaoming Guo and Pavel Zorin-Kranich.<\/p>\n\t\t","protected":false},"excerpt":{"rendered":"<p>The HADES seminar on Tuesday,&nbsp;October 5th,&nbsp;will be given by&nbsp;Ruixiang Zhang at&nbsp;5 pm&nbsp;in&nbsp;740 Evans. Speaker: Ruixiang Zhang (UC Berkeley) Abstract: Given a polynomial $P$ of constant degree in $d$ variables and consider the oscillatory integral $$I_P = \\int_{[0,1]^d} e(P(\\xi)) \\mathrm{d}\\xi.$$ Assuming $d$ is also fixed, what is a good upper bound of $|I_P|$? In this talk, [&hellip;]<\/p>\n","protected":false},"author":90,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-392","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/users\/90"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/comments?post=392"}],"version-history":[{"count":0,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/posts\/392\/revisions"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/media?parent=392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/categories?post=392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/hades\/wp-json\/wp\/v2\/tags?post=392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}