{"id":418,"date":"2025-12-09T20:26:36","date_gmt":"2025-12-09T13:26:36","guid":{"rendered":"https:\/\/algebra.math-ugm.id\/?p=418"},"modified":"2025-12-09T20:26:53","modified_gmt":"2025-12-09T13:26:53","slug":"demomatika-15-division-algebra-and-brauer-group-a-tour-through-theory-and-applications","status":"publish","type":"post","link":"https:\/\/algebra.math-ugm.id\/en\/demomatika-15-division-algebra-and-brauer-group-a-tour-through-theory-and-applications\/","title":{"rendered":"Demomatika #15: Division Algebra and Brauer Group: A tour through theory and applications"},"content":{"rendered":"

<\/p>\n

Speaker:<\/strong> Prof. Bharath Sethuraman<\/em> \u2013 California State University, Northridge<\/span>
Moderator:<\/strong> Dy Outdom<\/em><\/span>
Date:<\/strong> 3 December 2025<\/span>
Time:<\/strong> 09.30 \u2013 12.30 WIB<\/span>
Venue:<\/strong> Conference Room 1, 3rd Floor & Zoom Meeting<\/span><\/h2>\n

The Algebra Laboratory of Universitas Gadjah Mada hosted the second session of the 15th DEMOMATIKA Seminar Series<\/strong>, featuring a special lecture by Prof. Bharath Sethuraman<\/strong> on the topic \u201cDivision Algebra and Brauer Group: A Tour Through Theory and Applications.\u201d<\/strong> The lecture brought together faculty, students, and researchers interested in advanced topics in algebra, number theory, and their modern applications.<\/p>\n

\"\"\"\"<\/p>\n

Prof. Sethuraman began by introducing the fundamental notion of a division ring<\/strong>, illustrating the concept through the classical example of Hamilton\u2019s quaternions<\/strong>.<\/p>\n

\"\"<\/p>\n

He then presented further generalizations, in<\/span>cluding <\/span>quaternion algebras<\/strong> and <\/span>cyclic algebras<\/strong>, explaining how these structures arise naturally within the broader theory of division algebras. One of the key highlights was the application of division algebra theory to <\/span>space\u2013time coding<\/strong>, a technique used in wireless communication to increase reliability and performance. Prof. Sethuraman demonstrated how algebraic constructions lead to optimal coding schemes with strong theoretical foundations.<\/span>\"\"<\/p>\n

The lecture progressed to central simple algebras<\/strong> and the concept of crossed products<\/strong>, culminating in the statement of an important open question in the field: Is every division algebra of dimension p^2 <\/span><\/span>over its center\u00a0 a crossed product?<\/em><\/p>\n

\"\"<\/p>\n

 <\/p>\n

The problem remains unresolvedfor most primes, with affir<\/span>mative answers known only for small prim<\/span>es. Prof. Sethuraman then introduced the <\/span>Brauer group of a field<\/strong>, discussing its defining properties and illustrating key examples, including:<\/span><\/p>\n

    \n
  • \n

    Algebraically closed fields<\/p>\n<\/li>\n

  • \n

    Finite fields<\/p>\n<\/li>\n

  • \n

    p<\/span><\/span><\/span><\/span>-adic fields<\/p>\n<\/li>\n<\/ul>\n

    He also described how the Brauer group connects to group cohomology<\/strong> via natural homomorphisms, establishing deep links between algebra and topology. the lecture continued with the Brauer group of a commutative ring<\/strong>, also known as the theory of Azumaya algebras<\/strong>\u2014developed by Azumaya<\/strong> for the local case and by Auslander\u2013Goldman<\/strong> for the general case. Toward the end of the lecture, Prof. Sethuraman presented several major open problems currently driving research in division algebras and Brauer groups, including:<\/p>\n

      \n
    • \n

      Whether every division algebra of prime index<\/strong> (p\u22655<\/span><\/span>) is a crossed product<\/p>\n<\/li>\n

    • \n

      Whether every division algebra of prime exponent<\/strong> is a crossed product<\/p>\n<\/li>\n

    • \n

      The Period\u2013Index Conjecture<\/strong><\/p>\n

      If F<\/span><\/span> is a C^d<\/span><\/span>-field, then Br-dim(F)=d\u22121<\/span><\/p>\n

      \u00a0<\/span>\"\"<\/p>\n

      These problems remain at the forefront of modern research in noncommutative algebra and arithmetic geometry.<\/li>\n<\/ul>\n

      <\/p>","protected":false},"excerpt":{"rendered":"

      Speaker: Prof. Bharath Sethuraman \u2013 California State University, NorthridgeModerator: Dy […]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-418","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/418","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/comments?post=418"}],"version-history":[{"count":2,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/418\/revisions"}],"predecessor-version":[{"id":432,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/posts\/418\/revisions\/432"}],"wp:attachment":[{"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/media?parent=418"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/categories?post=418"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/algebra.math-ugm.id\/en\/wp-json\/wp\/v2\/tags?post=418"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}