{"id":197,"date":"2021-04-13T11:18:00","date_gmt":"2021-04-13T10:18:00","guid":{"rendered":"http:\/\/blogs.cardiff.ac.uk\/mathsugprojects\/?p=197"},"modified":"2022-11-11T10:24:55","modified_gmt":"2022-11-11T10:24:55","slug":"derived-categories-and-algebraic-geometry-dr-t-logvinenko","status":"publish","type":"post","link":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/derived-categories-and-algebraic-geometry-dr-t-logvinenko\/","title":{"rendered":"Derived Categories and Algebraic Geometry &#8211; Supervisor: Dr Timothy Logvinenko"},"content":{"rendered":"<p style=\"color: #383735\"><strong>Title of project:<\/strong><br \/>\nDerived Categories and Algebraic Geometry<\/p>\n<p style=\"color: #383735\"><strong>Code:<\/strong><br \/>\nTL2122B<\/p>\n<p style=\"color: #383735\"><strong>Supervisor:\u00a0<\/strong><br \/>\nDr Timothy Logvinenko<\/p>\n<p><strong>Project description:<\/strong><br \/>\nAlgebraic geometry studies geometrical objects by attaching invariants to them. For example any Riemann surface has its genus \u2014 the number of holes in it. More sophisticated examples include Betti and Hodge numbers, cohomology groups and ultimately \u2013 the derived category. The latter became over the last two decades the main technical tool of algebraic geometry, a sophisticated, abstract but very rewarding instrument.<\/p>\n<p>This project would first introduce students to the abstract notion of a derived category of an abelian category. The main example we want to study is the bounded derived category of vector bundles on a projective space. Ultimately, the students should learn to compute the derived categories of P^1 and P^2, one- and two-dimensional projective spaces.<\/p>\n<p>The project will involve guided reading, review of literature and writing up a report. There will be also be a minor computational element.<\/p>\n<p>Students who wish to take this project in Year 4 are strongly recommended to take this supervisor\u2019s Category Theory project in Year 3.<\/p>\n<p style=\"color: #383735\"><strong>Project offered as double module, single module, or both:<\/strong><br \/>\nDouble<\/p>\n<p style=\"color: #383735\"><strong>Prerequisite 3rd year modules:<\/strong><br \/>\nMA3013: Algebra II &#8211; Rings<br \/>\nMA0322: Algebra III &#8211; Fields<\/p>\n<p style=\"color: #383735\"><strong>Number of students who could be supervised for this project:<\/strong><br \/>\n1<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Title of project: Derived Categories and Algebraic Geometry Code: TL2122B Supervisor:\u00a0 Dr Timothy Logvinenko Project description: Algebraic geometry studies geometrical objects by attaching invariants to them. For example any Riemann surface has its genus \u2014 the number of holes in it. More sophisticated examples include Betti and Hodge numbers, cohomology groups and ultimately \u2013 the &hellip; <a href=\"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/derived-categories-and-algebraic-geometry-dr-t-logvinenko\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Derived Categories and Algebraic Geometry &#8211; Supervisor: Dr Timothy Logvinenko<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[2,4,8],"tags":[],"class_list":["post-197","post","type-post","status-publish","format-standard","hentry","category-2021-2022","category-double","category-yr-4-description-2021-2022"],"aioseo_notices":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/197","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/comments?post=197"}],"version-history":[{"count":4,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/197\/revisions"}],"predecessor-version":[{"id":2040,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/posts\/197\/revisions\/2040"}],"wp:attachment":[{"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/media?parent=197"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/categories?post=197"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.maths.cf.ac.uk\/mathsugprojects\/wp-json\/wp\/v2\/tags?post=197"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}