{"id":551,"global_id":"colleges.claremont.edu\/ccms?id=551","global_id_lineage":["colleges.claremont.edu\/ccms?id=551"],"author":"73","status":"publish","date":"2018-09-12 10:43:29","date_utc":"2018-09-12 17:43:29","modified":"2018-11-05 14:59:53","modified_utc":"2018-11-05 22:59:53","url":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-talk-by-yesim-demiroglu-hmc\/","rest_url":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/tribe\/events\/v1\/events\/551","title":"Cayley digraphs of matrix rings over finite fields (Yesim Demiroglu, HMC)","description":"<p>In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a sum of two SL_n-matrices when n&gt;1. We compute the eigenvalues of these graphs in terms of Kloosterman sums and study their spectral properties; and prove that if X is a subset of Mat_2 (F_q) with size |X| &gt; (2 q^3 \\sqrt{q})\/(q &#8211; 1), then X contains at least two distinct matrices whose difference has determinant $\\alpha$ for any $\\alpha \\in F_q^*$. Using this result we also prove a sum-product type result: if $A,B,C,D \\subseteq F_q$ satisfy $\\sqrt[4]{|A||B||C||D|}= \\Omega (q^{0.75})$ as q tends to infinity, then $(A &#8211; B)(C &#8211; D)$ equals all of $F_q$. In particular, if A is a subset of F_q with cardinality $|A| &gt; \\frac{3}{2} q^{3\/4}$, then the subset $(A &#8211; A) (A &#8211; A)$ equals all of $F_q$. We also recover a classical result: every element in any finite ring of odd order can be written as the sum of two units.\u00a0This talk should be accessible to undergraduates with some background in linear algebra.<\/p>","excerpt":"","slug":"antc-talk-by-yesim-demiroglu-hmc","image":false,"all_day":false,"start_date":"2018-11-13 12:15:00","start_date_details":{"year":"2018","month":"11","day":"13","hour":"12","minutes":"15","seconds":"00"},"end_date":"2018-11-13 13:10:00","end_date_details":{"year":"2018","month":"11","day":"13","hour":"13","minutes":"10","seconds":"00"},"utc_start_date":"2018-11-13 20:15:00","utc_start_date_details":{"year":"2018","month":"11","day":"13","hour":"20","minutes":"15","seconds":"00"},"utc_end_date":"2018-11-13 21:10:00","utc_end_date_details":{"year":"2018","month":"11","day":"13","hour":"21","minutes":"10","seconds":"00"},"timezone":"America\/Los_Angeles","timezone_abbr":"PST","cost":"","cost_details":{"currency_symbol":"","currency_code":"","currency_position":"prefix","values":[]},"website":"","show_map":true,"show_map_link":true,"hide_from_listings":false,"sticky":false,"featured":false,"categories":[{"name":"Algebra \/ Number Theory \/ Combinatorics Seminar","slug":"antc-seminar","term_group":0,"term_taxonomy_id":13,"taxonomy":"tribe_events_cat","description":"","parent":0,"count":171,"filter":"raw","term_order":"0","id":13,"urls":{"self":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/tribe\/events\/v1\/categories\/13","collection":"https:\/\/colleges.claremont.edu\/ccms\/wp-json\/tribe\/events\/v1\/categories"}}],"tags":[],"venue":{"id":425,"author":"73","status":"publish","date":"2018-08-17 08:08:12","date_utc":"2018-08-17 15:08:12","modified":"2018-09-26 17:19:34","modified_utc":"2018-09-27 00:19:34","url":"https:\/\/colleges.claremont.edu\/ccms\/venue\/millikan-2099-pomona-college\/","venue":"Millikan 2099, Pomona College","slug":"millikan-2099-pomona-college","address":"610 N. College Ave.","city":"Claremont","country":"United States","state":"CA","zip":"91711","stateprovince":"CA","json_ld":{"@type":"Place","name":"Millikan 2099, Pomona College","description":"","url":"https:\/\/colleges.claremont.edu\/ccms\/venue\/millikan-2099-pomona-college\/","address":{"@type":"PostalAddress","streetAddress":"610 N. College Ave.","addressLocality":"Claremont","addressRegion":"CA","postalCode":"91711","addressCountry":"United States"},"telephone":"","sameAs":""},"show_map":true,"show_map_link":true,"global_id":"colleges.claremont.edu\/ccms?id=425","global_id_lineage":["colleges.claremont.edu\/ccms?id=425"]},"organizer":[],"custom_fields":[],"json_ld":{"@context":"http:\/\/schema.org","@type":"Event","name":"Cayley digraphs of matrix rings over finite fields (Yesim Demiroglu, HMC)","description":"&lt;p&gt;In this talk we use the unit-graphs and the special unit-digraphs on matrix rings to show that every n x n nonzero matrix over F_q can be written as a [&hellip;]&lt;\/p&gt;\\n","url":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-talk-by-yesim-demiroglu-hmc\/","eventAttendanceMode":"https:\/\/schema.org\/OfflineEventAttendanceMode","eventStatus":"https:\/\/schema.org\/EventScheduled","startDate":"2018-11-13T12:15:00-08:00","endDate":"2018-11-13T13:10:00-08:00","location":{"@type":"Place","name":"Millikan 2099, Pomona College","description":"","url":"https:\/\/colleges.claremont.edu\/ccms\/venue\/millikan-2099-pomona-college\/","address":{"@type":"PostalAddress","streetAddress":"610 N. College Ave.","addressLocality":"Claremont","addressRegion":"CA","postalCode":"91711","addressCountry":"United States"},"telephone":"","sameAs":""},"performer":"Organization"},"ticketed":false,"is_virtual":false,"virtual_url":null,"virtual_video_source":""}