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mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:10.0pt; font-family:"Times New Roman","serif";} </style> <![endif]--> <link rel=stylesheet href=style.css type="text/css" media=screen> <!--[if gte mso 9]><xml> <o:shapedefaults v:ext="edit" spidmax="11266"/> </xml><![endif]--><!--[if gte mso 9]><xml> <o:shapelayout v:ext="edit"> <o:idmap v:ext="edit" data="1"/> </o:shapelayout></xml><![endif]--> </head> <body lang=EN-US link=blue vlink=purple style='tab-interval:.5in'> <div class=WordSection1> <p class=MsoNormal align=center style='text-align:center'><b style='mso-bidi-font-weight: normal'><span style='font-size:20.0pt'>Higher Order Fourier <span class=GramE>Analysis</span><o:p></o:p></span></b></p> <p class=MsoNormal align=center style='text-align:center'><o:p>&nbsp;</o:p></p> <p class=MsoNormal align=center style='text-align:center'>Summer school</p> <p class=MsoNormal align=center style='text-align:center'><span class=SpellE>Alfr</span><span lang=HU style='mso-ansi-language:HU'>é</span>d <span class=SpellE>Rényi</span> Institute of Mathematics, Budapest </p> <p class=MsoNormal align=center style='text-align:center'>June 2-4, 2011</p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <div class=MsoNormal align=center style='text-align:center'><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman";mso-ansi-language:EN-AU'> <hr size=3 width="100%" align=center> </span></div> <div id=content> <p class=MsoNormal><span class=GramE><span class=contenttitle><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman";mso-ansi-language:EN-AU'>Announcement.</span></span></span><span class=contenttitle><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'> There will be a Summer School on </span></span>Higher Order Fourier Analysis <span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>in the </span><span class=SpellE>Alfr</span><span lang=HU style='mso-ansi-language:HU'>é</span>d <span class=SpellE>Rényi</span> Institute of Mathematics <span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>on June 2-4, 2011. <br> <br> The School will <span class=SpellE>center</span> on a series of lectures given by <span class=SpellE>Balázs</span> <span class=SpellE>Szegedy</span> (University of Toronto). </span>In his seminal paper Timothy <span class=SpellE>Gowers</span> introduced a sequence of norms <span class=GramE>U(</span>k) defined for functions on <span class=SpellE>Abelian</span> groups, later named as <span class=SpellE>Gowers</span> norms. He used these norms to give quantitative bounds for <span class=SpellE>Szemeredi's</span> theorem on arithmetic progressions. The behavior of the <span class=GramE>U(</span>2) norm is closely tied to classical Fourier analysis, but it was unclear what would be the analogous notion for higher norms. The speaker will present a generalization of Fourier analysis (called higher order Fourier analysis) that solves this problem in a meaningful way. Ordinary Fourier analysis deals with <span class=SpellE>homomorphisms</span> of <span class=SpellE>Abelian</span> groups into the circle group; k-<span class=SpellE>th</span> order Fourier analysis deals with <span class=SpellE>morphisms</span> of <span class=SpellE>Abelian</span> groups into algebraic structures called `k-step <span class=SpellE>nilspaces</span> . These structures are variants of the parallelepiped structures previously introduced by Host and <span class=SpellE>Kra</span> and they are close relatives of <span class=SpellE>nilmanifolds</span>. The new approach presented by <span class=SpellE>Szegedy</span> has two main components: an underlying algebraic theory of <span class=SpellE>nilspaces</span> and a variant of <span class=SpellE>ergodic</span> theory on <span class=SpellE>ultraproduct</span> groups. Using this theory, one can obtain inverse theorems for the <span class=GramE>U(</span>k) norms on arbitrary <span class=SpellE>Abelian</span> groups, that generalize results by Green, Tao and Ziegler. This approach leads to a limit theory for functions on <span class=SpellE>Abelian</span> groups in the spirit of the recently developed theory of graph limits.&nbsp;</p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'><br> There is no registration fee for this school. If you plan to attend or have any questions, please contact one of the organizers, <span class=SpellE>Miklós</span> <span class=SpellE>Abért</span> (<span class=SpellE>abert</span> at <span class=SpellE>renyi</span> dot <span class=SpellE>hu</span>) or <span class=SpellE>László</span> <span class=SpellE>Lovász</span> (<span class=SpellE>lovasz</span> at <span class=SpellE>cs</span> dot <span class=SpellE>elte</span> dot <span class=SpellE>hu</span>). <o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span class=GramE><span lang=EN-AU style='mso-fareast-font-family: "Times New Roman";mso-ansi-language:EN-AU'>SCHEDULE OF LECTURES AND DISCUSSION SESSIONS.</span></span><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'> <o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>June 2, Thursday: 10.00  10.50 lecture, 11.05  11.55 lecture<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>June 3, Friday: 10.00  10.50 lecture, 11.05  11.55 lecture, 16.00  18.00 discussion session<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>June 4, Saturday: 10.00  10.50 lecture, 11.05  11.55 lecture, 14.00  16.00 discussion session<o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'>Coffee and tea will be available between the lectures. In the first lecture (Thursday 10.00  10.50) the speaker will present a broad overview of the subject. <o:p></o:p></span></p> <p class=MsoNormal><span lang=EN-AU style='mso-fareast-font-family:"Times New Roman"; mso-ansi-language:EN-AU'><br> </span><span class=GramE>Remark.</span> The school has an overlap of dates with the Paul <span class=SpellE>Turan</span> Memorial Lectures given by Yuval Peres (<span class=SpellE><span class=GramE>june</span></span> 1  3), but there will be no conflict of program. </p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <p class=MsoNormal>SOME LITERATURE FOR THE LECTURES: </p> <p class=MsoNormal>B. <span class=SpellE>Szegedy</span>: &quot;<span class=SpellE>Gowers</span> norms, regularization and limits of functions on <span class=SpellE>abelian</span> groups&quot; <a href="http://arxiv.org/abs/1010.6211" target="_blank">http://arxiv.org/abs/1010.6211</a><br> B. <span class=SpellE>Szegedy</span>: &quot;<span class=SpellE>Nilspaces</span>, <span class=SpellE>nilmanifolds</span> and their <span class=SpellE>morphisms</span>&quot; <a href="http://arxiv.org/abs/1009.3825" target="_blank">http://arxiv.org/abs/1009.3825</a><br> B. <span class=SpellE>Szegedy</span>: &quot;Higher order Fourier analysis as an algebraic theory II.&quot; <a href="http://arxiv.org/abs/0911.1157" target="_blank">http://arxiv.org/abs/0911.1157</a><br> B. Host, B. <span class=SpellE>Kra</span>: &quot;Parallelepipeds, nilpotent groups and <span class=SpellE>Gowers</span> norms&quot; <a href="http://www.math.northwestern.edu/%7Ekra/papers/nil.pdf" target="_blank">http://www.math.northwestern.<wbr>edu/~kra/papers/nil.pdf</a><br> B. Green, T. Tao, T. Ziegler: &quot;An inverse theorem for the <span class=SpellE>Gowers</span> U^{s+1}(N)-norm&quot; <a href="http://arxiv.org/abs/1009.3998" target="_blank">http://arxiv.org/abs/1009.3998</a><br> T. <span class=SpellE>Gowers</span>: &quot;A new proof of <span class=SpellE>Szemeredi's</span> theorem&quot; <a href="http://www.cs.umd.edu/%7Egasarch/vdw/sz-thm-gowers-proof.pdf" target="_blank">http://www.cs.umd.edu/~<wbr>gasarch/vdw/sz-thm-gowers-<wbr>proof.pdf</a></p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <p class=MsoNormal>A SHORT DICTIONARY</p> <p class=MsoNormal><a href="algebraic.pdf">Here</a> is a short text that was distributed among the participants. </p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> </div> </div> </body> </html>