{"id":29,"date":"2018-05-17T14:25:33","date_gmt":"2018-05-17T18:25:33","guid":{"rendered":"http:\/\/casfaculty.case.edu\/nick-gurski\/?page_id=29"},"modified":"2018-05-18T15:25:24","modified_gmt":"2018-05-18T19:25:24","slug":"research","status":"publish","type":"page","link":"https:\/\/casfaculty.case.edu\/nick-gurski\/research\/","title":{"rendered":"Research"},"content":{"rendered":"<p><img loading=\"lazy\" class=\"aligncenter wp-image-42 size-medium\" src=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18150525\/shh2_diagram-300x172.png\" alt=\"\" width=\"300\" height=\"172\" srcset=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18150525\/shh2_diagram-300x172.png 300w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18150525\/shh2_diagram-500x286.png 500w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18150525\/shh2_diagram.png 538w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2>What is category theory?<\/h2>\n<p>Category theory can be succinctly defined as the study of things and the relationships between them: we don&#8217;t really care so much\u00a0<em>which<\/em> things, as long as the relationships follow some basic rules. Examples include numbers and the relationship\u00a0<em>x is less than or equal to y<\/em>, points on a map and the relationship\u00a0<em>you can walk from point A to point B<\/em>, or sets and the relationships given by\u00a0<em>functions between those sets<\/em>. Category theory got its start by providing a necessary language for the development of algebraic topology, but since then has greatly expanded in scope and influence.<\/p>\n<h2>What do I actually do?<\/h2>\n<p>I started off doing what you might call hardcore 3-dimensional category theory; the 3 here means you study three different layers of relationships instead of just the usual one. As time has gone on, I have gotten more interested in topological, algebraic, and physical applications of this theory. I am generally interested in extra algebraic structure on categorical objects, such as monads, operads, or distinguished dualizable or invertible objects\/higher cells. Current research projects include<\/p>\n<ul>\n<li>studying how Picard 2-categories encode the algebraic information in stable homotopy 2-types, with a particular emphasis on translating topological invariants into categorical structure;<\/li>\n<li>an exploration of the interaction between classical operads and group actions, with an application to invertible objects in different kinds of monoidal categories; and<\/li>\n<li>an analogue for quasicategories of Beck&#8217;s theorem on distributive laws, using the homotopy coherent monads of Riehl-Verity.<\/li>\n<\/ul>\n<p>I&#8217;d like to get more into TQFTs and higher categories of modules.<\/p>\n<h2>Publications<\/h2>\n<p>You can find a full list of my publications\u00a0<a href=\"http:\/\/casfaculty.case.edu\/nick-gurski\/publications\/\">here<\/a>.<\/p>\n<h2>And what do my students do?<\/h2>\n<p>My past PhD students, along with thesis titles, are:<\/p>\n<ul>\n<li>Thomas Athorne,\u00a0<em>Coalgebraic cell complexes<\/em><\/li>\n<li>Alexander Corner,\u00a0<em>Day convolution for monoidal bicategories<\/em><\/li>\n<li>Edward Prior,\u00a0<em>Action operads and the free G-monoidal category on n invertible objects.<\/em><\/li>\n<\/ul>\n<p>Some other possible projects for future students include:<\/p>\n<ul>\n<li>prove coherence for lax homomorphisms between tricategories or monoidal 2-categories of some type,<\/li>\n<li>investigate Leinster-style operads using profunctors instead of spans,<\/li>\n<li>investigate the 2-categorical aspects of objects like spherical or modular categories, and<\/li>\n<li>prove the Homotopy Hypothesis for iterated, weakly enriched categories.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-44\" src=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math-1024x723.png\" alt=\"\" width=\"452\" height=\"319\" srcset=\"https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math-1024x723.png 1024w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math-300x212.png 300w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math-768x542.png 768w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math-500x353.png 500w, https:\/\/artscimedia.case.edu\/wp-content\/uploads\/sites\/216\/2018\/05\/18151538\/fail_math.png 1393w\" sizes=\"(max-width: 452px) 100vw, 452px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>What is category theory?<br \/>\nCategory theory can be succinctly defined as the study of things and the relationships between them: we don&#8217;t really care so much\u00a0<em>which<\/em> things, as long as the relationships follow some basic rules. Examples include numbers and the relationship\u00a0<em>x is less than or equal to y<\/em>, points on a map and the relationship\u00a0<em>you can walk from point A to point B<\/em>, or sets and the relationships given by\u00a0<em>functions between those sets<\/em>. Category theory got its start by providing a necessary language for the development of algebraic topology,<\/p>\n<p><a href=\"https:\/\/casfaculty.case.edu\/nick-gurski\/research\/\" class=\"more-link\">Continue reading&#8230; <span class=\"screen-reader-text\">Research<\/span><\/a><\/p>\n","protected":false},"author":298,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/pages\/29"}],"collection":[{"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/users\/298"}],"replies":[{"embeddable":true,"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/comments?post=29"}],"version-history":[{"count":6,"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/pages\/29\/revisions"}],"predecessor-version":[{"id":46,"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/pages\/29\/revisions\/46"}],"wp:attachment":[{"href":"https:\/\/casfaculty.case.edu\/nick-gurski\/wp-json\/wp\/v2\/media?parent=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}