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Creating Connections.</h1> <p class="catchphrase">The American Mathematical Society is dedicated to advancing research and connecting the diverse global mathematical community through publications, meetings and conferences, MathSciNet, professional services, advocacy, and awareness programs.</p> <p><a class="btn btn-info" href="/membership" role="button">Join us today »</a></p> </header>  <main> <section id="uppertier"> <ul class="nav nav-tabs" id="homepageTabs" role="tablist"> <li class="h4 active" role="presentation"><a aria-controls="tab-research" data-target="#researchTab" data-toggle="tab" href="#tab-research" role="tab">Research</a></li> <li class="h4" role="presentation"><a aria-controls="tab-news" data-target="#newsTab" data-toggle="tab" href="#tab-news" role="tab">News</a></li> <li class="h4" role="presentation"><a aria-controls="tab-calendar" data-target="#calendarTab" data-toggle="tab" href="#tab-calendar" role="tab">Events</a></li> <li class="h4" role="presentation"><a aria-controls="tab-deadlines" data-target="#deadlinesTab" data-toggle="tab" href="#tab-deadlines" role="tab">Deadlines</a></li> </ul>  <div class="tab-content"> <div class="tab-pane fade in active" id="researchTab" role="tabpanel"> <div> <article class="row"> <p class="sectionhead">From Notices of the AMS</p> <h2><a href="https://www.ams.org/journals/notices/202411/noti3080/noti3080.html?adat=December%202024&trk=3080&pdfissue=202411&pdffile=rnoti-p1471.pdf&cat=none&type=.html">Foliations and Diffeomorphism Groups</a></h2> <figure class="pull-right" style="max-width: 300px;"><img alt="artistic rendering of John Mather, William Thurston, and Andre Haefliger and shapes related to their work" class="img-responsive" src="/images/cover-cmyk-300x200.jpg" style="margin-left:5px" /></figure> <p class="AMS_gray_foreground small">by <a href="https://mathscinet.ams.org/mathscinet/mrauthorid/1228088">Sam Nariman</a><br /> <em>Communicated by <a href="https://mathscinet.ams.org/mathscinet/mrauthorid/641800">Chikako Mese</a></em></p> <p><strong>Introduction</strong></p> <p>Thurston, at his ICM address in 1974, asked "Given a large supply of some sort of fabrics, what kind of manifolds can be made from it, in a way that the patterns match up along the seams?" And he added that, "For open manifolds, Gromov's theorem gives a good answer for a wide variety of fabrics. The techniques needed to analyze such a question on a closed manifold are usually different, at least to a casual eye." In this article, we shall focus on one aspect of this difference between closed and open manifolds that is inspired by a property of diffeomorphism groups called <em>the fragmentation property</em>.</p> <p>But let us first define the basic objects that we would like to study. Recall that manifolds are modeled locally on Euclidean spaces $\mathbb{R}^n$ along with a subgroup of the homeomorphism group $\operatorname{Homeo}(\mathbb{R}^n)$ as the permissible transition functions for a given structure on the manifold. Now a $C^r$-<em>foliation</em> of codimension $k$ on an $n$-dimensional manifold $M$ is determined by a <em>foliated atlas</em> $\{(U_i, \phi_i)\}$ where $\{U_i\}$ is an open cover of $M$ and each $\phi_i\colon U_i\to \mathbb{R}^n$ is a homeomorphism onto an open set in $\mathbb{R}^n$ such that for overlapping pairs $U_i$ and $U_j$, the transition function $\phi_j\circ \phi_i^{-1}$ is a $C^r$ diffeomorphism from $\phi_i(U_i\cap U_j)$ to $\phi_j(U_i\cap U_j)$ of the form...</p> <p><small><a href="https://www.ams.org/journals/notices/202411/noti3080/noti3080.html?adat=December%202024&trk=3080&pdfissue=202411&pdffile=rnoti-p1471.pdf&cat=none&type=.html">Read more»</a></small></p> <dl> <dt><strong>Also in Notices</strong></dt> <dd><a href="https://www.ams.org/journals/notices/202411/noti3079/noti3079.html?adat=December%202024&trk=3079&pdfissue=202411&pdffile=rnoti-p1458.pdf&cat=none&type=.html">Connections Between Fractal Geometry and Projections</a></dd> <dd><a href="https://www.ams.org/journals/notices/202411/noti3083/noti3083.html?adat=December%202024&trk=3083&pdfissue=202411&pdffile=rnoti-p1534.pdf&cat=none&type=.html">What is... Optical Geometry?</a></dd> </dl> <div class="clearfix"> </div> <p class="pull-right"><a class="btn btn-default" href="https://www.ams.org/notices" role="button">More from Notices»</a></p> </article> </div> </div> <div class="tab-pane" id="newsTab" role="tabpanel"> <div> <div class="clearfix"> </div> <article class="col-md-2 hidden-sm hidden-xs" id="story1"> <a href="https://www.ams.org/news?class_id=1&expand=1&type=news"><img src="/images/homepage-news-image.png" alt="AMS News" width="318" height="318" class="img-responsive hidden-sm hidden-md" style="margin-top:20px;" /></a> </article> <article class="col-md-5" id="story2"> <header> <h2 class="AMS_orange_foreground">News from the AMS</h2> </header> <script src="/js/ajax.js"></script> <script src="/content/js/news_items.js"></script> <article id="amsNews1"></article> <script> var amsNews1; function amsNews1Load () { amsNews1 = new amsNews(); amsNews1.newsObject = "amsNews1"; amsNews1.newsClassID = 1; amsNews1.newsLimitCount = 4; 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