Computational Algebra

Traditionally, algebraists have been concerned with building theories that attempt to classify the structures satisfying a particular set of axioms. A well-known example is Cartan's classification of all simple Lie algebras over the field of complex numbers. However, over the past 20 years there has been a growing realization that many, if not all, branches of algebra have a rich algorithmic theory, in the sense that it is possible to design powerful algorithms capable of answering a great variety of questions about particular algebraic structures.

The design of effective algorithms, while building on standard theoretical results, often involves the introduction of new concepts and may utilize machinery from quite different branches of mathematics. For example, the introduction of elliptic curve methods led to a great improvement in techniques for factoring integers. Efficient algorithms for algebraic problems find wide application both in direct applications of algebra (e.g. in cryptography, coding theory, digital signal processing, experimental design, robotics) and as fundamental building blocks in algorithms designed to find exact solutions of differential equations. The interests of the research group span the areas of algebra, number theory and geometry (both algebraic and finite).

• Group Theory
  finitely presented groups, permutation groups, matrix groups, soluble groups, Lie groups, representation theory;
• Number Theory
  finite fields, factorization of integers, primality testing, local and global fields, Artin rings, Galois theory;
• Linear Algebra
  linear algebra over PIRs, canonical forms, structure of R-modules, homological algebra;
• Commutative Algebra
  polynomial factorization, Groebner basis methods, constructive ideal theory;
• Algebras
  associative algebras, division algebras, group algebras, Lie algebras;
• Algebraic Geometry
  plane curves, surfaces, general varieties, schemes;
• Finite Geometry
  designs, codes, graphs, geometries;
• Algebraic Programming Languages
  models of mathematical computation, language design, semantics, implementation techniques, user interfaces.
• Applications
  cryptography, coding theory, signal processing,quantum computation.

Professor J. J. Cannon (J.Cannon@maths.usyd.edu.au)