{"id":1867,"date":"2022-10-02T16:47:51","date_gmt":"2022-10-02T15:47:51","guid":{"rendered":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/?p=1867"},"modified":"2024-04-11T17:37:15","modified_gmt":"2024-04-11T16:37:15","slug":"phd-course-on-lattice-ordered-groups-and-polyhedral-geometry-spring-2023","status":"publish","type":"post","link":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/1867-phd-course-on-lattice-ordered-groups-and-polyhedral-geometry-spring-2023\/","title":{"rendered":"PhD course on lattice-ordered groups and polyhedral geometry (Spring 2023)"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\" id=\"introduction\">Introduction<\/h2>\n\n\n\n<p>The course is an introduction to the theory of abelian lattice-ordered groups from different perspectives. Initially, we study these structures with purely algebraic methods. We will analyse some important theorems and connections with other parts of mathematics, such as AF C*-algebras. Later we will move on to their geometric study, through the Baker-Beynon duality. It will be seen that, just as the commutative rings provide an algebraic counterpart for the study of affine manifolds with polynomial maps, lattice-ordered groups represent the algebraic counterpart of the polyhedral cones and piece-wise linear homogenous maps between them.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"course-topics\">Course topics<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Abelian lattice-ordered groups: definition and examples.<\/li>\n\n\n\n<li>Representation results.<\/li>\n\n\n\n<li>Archimedeanity and strong (order) unit.<\/li>\n\n\n\n<li>Free and finitely presented abelian l-groups.<\/li>\n\n\n\n<li>Baker&amp;Beynon duality.<\/li>\n\n\n\n<li>Polyhedral geometry<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"lesson-topics\">Lecture by lecture topics<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<ul class=\"wp-block-list\">\n<li>5\/5\/2023: Introduction to the course.  Motivations and applications of the theory of abelian lattice ordered groups. Main examples. Crash course on Galois connections and categorical adjunctions.<\/li>\n\n\n\n<li>11\/5\/2023: Overview of the main results and techniques in the study of l-groups: The integers and Weinberg&#8217;s theorem. Archimedeanicity and H\u00f6lder&#8217;s theorem. Semisimplicity and Yosida&#8217;s representation.<\/li>\n\n\n\n<li>12\/5\/2023: Strong unit and MV-algebras. The free abelian l-groups as algebras of functions. Lattice ordered groups and piecewise linear geometry. <\/li>\n\n\n\n<li>18\/5\/2023: The general adjunction and its fixed points: Baker&amp;Beynon duality and polyhedral geometry.<\/li>\n<\/ul>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"course-material\">Course material<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Bigard, A., Keimel, K., &amp; Wolfenstein, S. (2006). Groupes et anneaux r\u00e9ticul\u00e9s (Vol. 608). Springer.<\/li>\n\n\n\n<li>Anderson, M. E., &amp; Feil, T. H. (2012). Lattice-ordered groups: an introduction (Vol. 4). Springer Science &amp; Business Media.<\/li>\n\n\n\n<li>Goodearl, K. R. (2010).&nbsp;<em>Partially ordered abelian groups with interpolation<\/em>&nbsp;(No. 20). American Mathematical Soc.<\/li>\n\n\n\n<li>Glass, A. M. W. (1999).&nbsp;<em>Partially ordered groups<\/em>&nbsp;(Vol. 7). World Scientific.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"practical-aspects\">Practical aspects<\/h2>\n\n\n\n<h2 class=\"wp-block-heading has-medium-font-size\">Term and schedule<\/h2>\n\n\n\n<p>Lecturer: Luca Spada<br>Course duration: 10 hours.<br>Course calendar: 5, 11, 12 and 18 of May, from&nbsp;10:00 to 12:45.  All lectures are in room P19 (last floor, building F3).<br><\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"exam\">Exam<\/h3>\n\n\n\n<p>You can choose to take the final exam in one of the following ways:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A short oral interview (about 30 minutes) in which the knowledge acquired on the basic and more advanced concepts will be evaluated.<\/li>\n\n\n\n<li>The presentation of a topic agreed with the teacher and not covered in the course, in the form of a short seminar also open to other doctoral students lasting about 30 minutes.<\/li>\n\n\n\n<li>Solving some exercises at home.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Introduction The course is an introduction to the theory of abelian lattice-ordered groups from different perspectives. Initially, we study these structures with purely algebraic methods. We will analyse some important theorems and connections with other parts of mathematics, such as AF C*-algebras. Later we will move on to their geometric study, through the Baker-Beynon duality. [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[225,226,55,223,228,227,229,224,230],"class_list":["post-1867","post","type-post","status-publish","format-standard","hentry","category-teaching","tag-baker","tag-beynon","tag-duality","tag-l-group","tag-piece-wise-linear-maps","tag-polyhedral-geometry","tag-riez-spaces","tag-strong-order-unit","tag-vector-lattices"],"blocksy_meta":[],"_links":{"self":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1867","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/comments?post=1867"}],"version-history":[{"count":10,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1867\/revisions"}],"predecessor-version":[{"id":2194,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/posts\/1867\/revisions\/2194"}],"wp:attachment":[{"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/media?parent=1867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/categories?post=1867"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/logica.dipmat.unisa.it\/lucaspada\/wp-json\/wp\/v2\/tags?post=1867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}