In the last 20 years, operator algebras has moved from being a branch of functional analysis to a central field in mathematics with links and applications to many branches of pure mathematics and theoretical physics including topology, geometry, K-theory on the pure mathematics side and statistical mechanics, quantum field theory, string theory and spectral theory on the physics side.

This course is an introduction to the mathematical side of this field – with emphasis on learning through examples rather than the most abstract general theorems. Hence prerequisites are not necessary beyond what is covered in the compulsory core modules.

• Understand the basic theory of operator algebras
• Appreciate the need for noncommutative geometry
• Understand basic concepts in noncommutative geometry
• Demonstrate some of the basic principles of noncommutative geometry through examples from a wide spectrum

30 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 120 hours private study including preparation of worked solutions for problem classes.

Skills:

Ability to understand, manipulate and formulate abstract mathematical concepts

Transferable Skills:

Critical thinking and problem solving skills. A

bility to present work in a scholarly manner

Formative assessment is carried out in the problem classes.  Feedback to students on their solutions and their progress towards learning outcomes is provided during these classes.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from five equally weighted questions.

• Finite Dimensional Operator Algebras
• Commutative Operator Algebras
• Toeplitz Algebra
• Orbifolds: folding the interval and circle
• AF and Cuntz algebras
• Noncommutative Torus: irrational rotations
• Dimension: nonintegral examples
• Homology and K-theory – introduced via above examples

Examples from:

• Mathematical physics, including applications to statistical mechanics,  aperiodic systems and quasi crystals, quantum Hall effect.

• Topology and Biochemistry introduced via problems in DNA

Quantum Symmetries on Operator Algebras, Evans, D. E. & Kawahigashi, Y., Oxford University Press, 1998.

Operator algebras and Quantum Statistical Mechanics (Volumes I and II), Bratteli, O & Robinson, D., Springer Verlag, 2^nd editions

C*-algebras by example, Davidson, K.