{"id":2876,"date":"2020-09-02T18:20:02","date_gmt":"2020-09-02T22:20:02","guid":{"rendered":"http:\/\/matroidunion.org\/?p=2876"},"modified":"2020-09-07T21:25:26","modified_gmt":"2020-09-08T01:25:26","slug":"online-talk-matt-baker","status":"publish","type":"post","link":"https:\/\/matroidunion.org\/?p=2876","title":{"rendered":"Online talk: Matt Baker"},"content":{"rendered":"\n<p><strong>Mon, September 7,<\/strong> 3pm ET (8pm BST, 7am Tue NZST)<br \/><a href=\"http:\/\/people.math.gatech.edu\/~mbaker\/\"><strong>Matt Baker<\/strong><\/a>, Georgia Tech<br \/><strong>Foundations of Matroids without Large Uniform Minors, Part 1<br \/><\/strong><a href=\"https:\/\/youtu.be\/ThxWphUJYX0\"><strong>Youtube<\/strong><\/a><b><\/b><\/p>\n<p><strong>Abstract:<br \/><\/strong>Matroid theorists are of course very interested in questions concerning representability of matroids over fields. More generally, one can ask about representability over <em>partial fields<\/em> in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the <em>universal partial field<\/em> of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, almost all matroids are not representable over any partial field, and in this case, the universal partial field gives no information.<\/p>\n<p>Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the <em>foundation<\/em> of a matroid. The foundation of $M$ is a type of algebraic object which we call a <strong>pasture<\/strong>; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid&#8217;s theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$.<\/p>\n<p>As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for matroids having no $U(2,5)$ or $U(3,5)$ minors. The proof of this classification theorem relies crucially on Tutte&#8217;s Homotopy Theorem and the theory of cross-ratios for matroids. Among other things, our classification provides a short conceptual proof of the 1997 theorem of Lee and Scobee which says that a matroid is both ternary and orientable if and only if it is dyadic.<\/p>\n<p>This is part 1 of a series of two talks. The second talk will be given the following week by Oliver Lorscheid.<\/p>\n<p><strong>\u00a0<\/strong><\/p>\n<div>\u00a0<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Mon, September 7, 3pm ET (8pm BST, 7am Tue NZST)Matt Baker, Georgia TechFoundations of Matroids without Large Uniform Minors, Part 1Youtube Abstract:Matroid theorists are of course very interested in questions concerning representability of matroids over fields. More generally, one can &hellip; <a href=\"https:\/\/matroidunion.org\/?p=2876\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[11],"class_list":["post-2876","post","type-post","status-publish","format-standard","hentry","category-matroids","tag-online-talks"],"_links":{"self":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/2876","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2876"}],"version-history":[{"count":3,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/2876\/revisions"}],"predecessor-version":[{"id":3024,"href":"https:\/\/matroidunion.org\/index.php?rest_route=\/wp\/v2\/posts\/2876\/revisions\/3024"}],"wp:attachment":[{"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2876"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2876"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matroidunion.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2876"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}