- Visiting Scientist (2003)
Computational Algebra Group
Computational Algebra Group
The basic mission of the Computational Algebra Group is the development of computer techniques for solving symbolic problems in mathematics, with particular emphasis on the areas of algebra, number theory and geometry. The activities extend from the design of algorithms through to the construction of computer algebra packages and languages. The group concentrates on what might be called discrete algebraic and combinatorial structures: groups, rings, fields, modules, algebras, designs and geometries. A particular strength of the group is its expertise in computational methods across several major branches of mathematics.
The Computational Algebra Group provides a unique environment in that it brings together mathematicians, computer scientists and software engineers. The development of good algorithms requires the involvement of leading researchers from the relevant areas of mathematics. The efficient implementation and embedding of these algorithms in large integrated software systems requires sophisticated techniques from computer science. The design and implementation of algebraic programming languages requires the involvement of both mathematicians and computer scientists.
The group holds a regular Computational Algebra Seminar, and hosts approximately twenty visitors each year. The Computational Algebra Group also distributes the computer algebra system Magma, which has been cited in approximately 4000 research publications.
Current Members
Professors
John Cannon
Research interests include computational methods in algebra, geometry, number theory and combinatorics; the design of mathematical programming languages and the integration of databases with Computer Algebra systems. Contributions include overall concept and planning, language design, specific design for many categories, numerous algorithms (especially in group theory) and general management.
John Voight
Implemented algorithms for quaternion algebras over number fields, associative orders (with Nicole Sutherland), and Shimura curves.
Senior Research Fellows
Allan Steel
Has developed many of the fundamental data structures and algorithms in Magma for multiprecision integers, finite fields, matrices and modules, polynomials and Groebner bases, aggregates, memory management, environmental features, and the package system, and has also worked on the Magma language interpreter. In collaboration, he has developed the code for lattice theory (with Bernd Souvignier), invariant theory (with Gregor Kemper) and module theory (with Jon Carlson and Derek Holt).
Andreas-Stephan Elsenhans
Main research interests are in the areas of arithmetic and algebraic geometry, particularly cubic and K3 surfaces. Main contributions focus on cubic surfaces from the arithmetic and algebraic points of view.
Bill Unger
Main area of interest is computational group theory, with particular emphasis on algorithms for permutation and matrix groups. Implemented many of the current permutation and matrix group algorithms for Magma, in particular BSGS verification, solvable radical and chief series algorithms. Recently discovered a new method for computing the character table of a finite group.
Don Taylor
Research interests are in reflection groups, finite group theory, and geometry. Implemented algorithms for complex reflection groups and complex root data. Contributed to the packages for Chevalley groups and groups of Lie type. Currently developing algorithms for classical groups of isometries, Clifford algebras and spin groups.
Research Fellows
Geoff Bailey
Main interests include elliptic curves (especially those defined over the rationals), virtual machines and computer language design. Has implemented part of the elliptic curve facilities especially the calculation of Mordell--Weil groups. Other main areas of contribution include combinatorics, local fields and the Magma system internals.
Rhys J. Evans
Research Associate
Reymond Akpanya
Research Affiliates
Edgar Costa
Eran Assaf
Collaborating Member
Andrew V. Sutherland
Research Assistant
Nicole Sutherland
Works in the areas of arithmetic fields and algebraic geometry. Developed the machinery for Newton polygons and lazy power series and contributed to the code for local fields, number fields, modules over Dedekind domains, function fields, schemes and has worked on aspects of algebras.
Postdoctoral Research Fellow
Lewis Combes
PhD Student
Blair Butler
Research Administration Officer
Priscilla Yin-Ting Yuen
Honorary Research Associate
Mark Watkins
Works in the area of number theory, particularly analytic methods for arithmetic objects. Implemented a range of analytic tools for the study of elliptic curves including analytic rank, modular degree, Heegner points and (general) point searching methods. Also deals with conics, lattices, modular forms, and descent machinery over the rationals.
Historical Members
All members of the Computational Algebra Group.
- Research Fellow (2009–2010) From Nov 1
- Member (1995–1998)
- PhD Student (2012–2018)
- Computer Systems Officer (1992–1995)
- Senior Research Assistant (1995–2000)
- Senior Research Associate (2001–2002)
- PhD Student (2001–2011) P/T
- Research Fellow (2003–2009)
- Senior Research Fellow (2010–present)
- Visiting Scientist (2007–2008) From 1 Oct; To Feb 28
- Research Fellow (2012–2015) From November 1; To March 2015
- Senior Research Fellow (2024–present)
- Collaborating Member (2024–present)
- Administrative Officer (1992–2025)
- Honorary Research Associate (2000–2008)
- Software Engineer (2000–2003)
- Member (1996–1997)
- Research Fellow (1998–2009)
- Senior Research Fellow (2010–present)
- PhD Student (2025–present)
- Research Associate (2011–2013) From April 1 to October 11
- Member (1990–1998)
- Software Engineer (2000–2002)
- Member (1989–1996)
- MSc Student (2008)
- MSc Student (2010)
- Postdoc (2000–2001) From April 1
- Research Fellow (2002–2007)
- Senior Research Fellow (2008–2011)
- Honorary Research Associate (2012–2026)
- Member (2001–2003)
- Software Engineer (2000–2006) To July 30
- Postdoc (2006)
- Visiting Scientist (2008–2010) ENS, Lyon, From July 1; ENS, Lyon
- Postdoc (2006) From 1 May
- Research Fellow (2010–2012) From May 1; To March 31
- Research Associate (2012–2014) From July 8; To July 2014
- Research Fellow (1999–2001)
- Senior Research Fellow (2010–present)
- Research Affiliate (2024–present)
- Member (1999)
- Research Affiliate (2024–present)
- Postdoc (1999–2001)
- Postdoc (2006–2007) From 1 Sept; To Feb 28
- PhD Student (2008–2015)
- Postdoc (1998–2000)
- Computer Systems Officer (1995–1998)
- Research Associate (1999–2009)
- Research Fellow (2010–present)
- Member (1989–1993)
- Member (2000–2001)
- Software Engineer (2002–2006)
- PhD Student (2003–2006)
- PhD Student (2008–2012) Cotuelle
- Member (1998–1999)
- PhD Student (2008–2011) Cotuelle
- Postdoc (2005–2007) From 1 Dec
- Postdoc (2010–2011) From Oct 14; to Mar 14
- Reader (1986–1999)
- Professor and Head (2000–2024)
- Professor (2024–present)
- Postdoc (2005–2006)
- Professor (2024–present)
- PhD Student (1999–2001) TU, Berlin
- Postdoctoral Research Fellow (2024–present)
- PhD Student (2008–2010)
- Research Associate (2022–2025)
- PhD Student (2015)
- Postdoc (2004–2005) From July 1; To June 30
- Research Fellow (2008–2020) From July 1
- Honorary Research Associate (2020–present)
- Postdoc (2009–2010) From 1 Oct
- Administrative Officer (2009–2013) From Sept 1; To November 29
- Postdoc (2003–2005) From July 1; To June 30
- Research Fellow (2005–2015) From July 1; To May 2015
- Research Fellow (2018–2020)
- Member (1987)
- Research Assistant (1999–present)
- PhD Student (2008–2015) P/T to April 2015
- Research Fellow (2002–2003) From Sept 1; To Sept 30
- Visiting Scientist (2002–2003)
- Software Engineer (1999–2000)
- Research Associate (2001–2003)
- Research Administration Officer (2026–present)
- Research Associate (2025–present)
- Research Fellow (2025–present)
- Member (1997–1998)
- Research Fellow (2000–2001) To Feb 1
- Postdoc (2000–2007) From Sept 9; USYD Post-doc, To July 31
- Research Fellow (2007–2009) From Aug 1
- Honorary Research Associate (2010–2018)
- Postdoc (2006–2007) To May 15
- Research Fellow (2005–2017) From Nov 1
- Postdoc (1999–2001) To Aug 31
- Research Fellow (2001–2003) From Sept 1; To Sept 1
- Honorary Research Associate (2003–2026)
- Member (1989–1996)
- Postdoc (2004–2005)
- PhD Student (2009–2015)