The Algebra Research Group

Speaker: Prof. Bharath Sethuraman – California State University, Northridge
Moderator: Dy Outdom
Date: 3 December 2025
Time: 09.30 – 12.30 WIB
Venue: Conference Room 1, 3rd Floor & Zoom Meeting

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

Prof. Sethuraman began by introducing the fundamental notion of a division ring, illustrating the concept through the classical example of Hamilton’s quaternions.

He then presented further generalizations, including quaternion algebras and cyclic algebras, 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 space–time coding, 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.

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

The problem remains unresolvedfor most primes, with affirmative answers known only for small primes. Prof. Sethuraman then introduced the Brauer group of a field, discussing its defining properties and illustrating key examples, including:

• Algebraically closed fields

• Finite fields

• $p$-adic fields

He also described how the Brauer group connects to group cohomology via natural homomorphisms, establishing deep links between algebra and topology. the lecture continued with the Brauer group of a commutative ring, also known as the theory of Azumaya algebras—developed by Azumaya for the local case and by Auslander–Goldman 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:

• Whether every division algebra of prime index (p≥5) is a crossed product

• Whether every division algebra of prime exponent is a crossed product

• The Period–Index Conjecture

  If F is a C^d-field, then Br-dim(F)=d−1

   

  These problems remain at the forefront of modern research in noncommutative algebra and arithmetic geometry.