{"id":48,"date":"2022-10-02T14:23:53","date_gmt":"2022-10-02T12:23:53","guid":{"rendered":"https:\/\/felix-schremmer.de\/?page_id=48"},"modified":"2024-05-10T04:12:42","modified_gmt":"2024-05-10T02:12:42","slug":"research-interests","status":"publish","type":"page","link":"https:\/\/felix-schremmer.de\/index.php\/research-interests\/","title":{"rendered":"Research Interests"},"content":{"rendered":"\n<p>A linear algebraic group <var>G<\/var> defined over a field <var>F<\/var> is, by definition, a group object in the category of affine <var>F<\/var>-varieties. As such, <var>G<\/var> can be studied using methods from algebraic geometry and group theory. \nIf <var>F<\/var> is a local field, e.g. the <var>p<\/var>-adic numbers, the geometry of <var>G<\/var> has number-theoretic significance. Such groups are used frequently in the Langlands program, a very influential research direction linking number theory with representation theory.<\/p>\n\n\n\n<p>My main object of interest are affine Deligne-Lusztig varieties. These are finite-dimensional schemes over the residue field of <var>F<\/var>, and are used primarily to study the special fibre of Rapoport-Zink moduli spaces. Via these Rapoport-Zink spaces, the geometry of affine Deligne-Lusztig varieties is related to Shimura varieties.<\/p>\n\n\n\n<p>The definition of affine Deligne-Lusztig varieties is motivated by a classical paper of Deligne and Lusztig. In a non-affine setting, they construct certain varieties, whose cohomology provides important representations of finite groups of Lie type. The geometry of these Deligne-Lusztig varieties is well-understood. This connection is not only motivational &#8211; sometimes the nice geometric properties of classical Deligne-Lusztig varieties come up when studying the affine case.<\/p>\n\n\n\n<p>On a more down-to-earth level, affine Deligne-Lusztig varieties compare two decompositions of the space <var>G(F\u02d8)<\/var>. The first one is the decomposition into \u03c3-conjugacy classes, a Galois twisted version of the usual conjugation in a group. The second one is the decomposition into double cosets <var>IxI<\/var>, where <var>I<\/var> is an Iwahori subgroup and <var>x\u2208G(F\u02d8)<\/var>. The intersection of such a \u03c3-conjugacy class <var>[b]<\/var> with an Iwahori double coset <var>IxI<\/var> is called Newton stratum, and its geometry is closely related to a corresponding affine Deligne-Lusztig variety <var>X<sub>x<\/sub>(b)<\/var>. Both decompositions are can be parametrized using an infinite Coxeter group, known as the extended Weyl group of <var>G<\/var>.<\/p>\n\n\n\n<p>The tools to study these affine Deligne-Lusztig varieties come from algebraic geometry, the theory of Coxeter groups, the theory of (affine) Lie algebras and (affine) root systems as well as quantum algebra. This makes affine Deligne-Lusztig varieties a very rich and dynamic research topic.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A linear algebraic group G defined over a field F is, by definition, a group object in the category of affine F-varieties. As such, G can be studied using methods from algebraic geometry and group theory. If F is a local field, e.g. the p-adic numbers, the geometry of G has number-theoretic significance. Such groups &#8230; <a title=\"Research Interests\" class=\"read-more\" href=\"https:\/\/felix-schremmer.de\/index.php\/research-interests\/\" aria-label=\"Read more about Research Interests\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/pages\/48"}],"collection":[{"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/comments?post=48"}],"version-history":[{"count":21,"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/pages\/48\/revisions"}],"predecessor-version":[{"id":127,"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/pages\/48\/revisions\/127"}],"wp:attachment":[{"href":"https:\/\/felix-schremmer.de\/index.php\/wp-json\/wp\/v2\/media?parent=48"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}