Professor Emeritus of Mathematics, University of California San Diego
J. William Helton’s group developed the package NCAlgebra for doing general-purpose noncommutative algebra in Mathematica. It began around 1990 and has been extended continually since then. From his work at the origins of noncommutative geometry and H-infinity control, Helton kept seeing such noncommutative formulas and hoping computer algebra could help. So, with Bob Miller, he started NCAlgebra and developed algorithms to find out. Around the year 2000, linear control theory shifted away from equalities to inequalities, e.g. from Riccati equations to linear matrix inequalities. This motivated Helton and a few others to begin what has developed into an elegant theory of noncommutative inequalities, to wit, a noncommutative version of real algebraic geometry. NCAlgebra seriously accelerated (and was accelerated by) this development.
A booming area full of noncommutative algebra is quantum information theory, and that is the main direction of current NCAlgebra development. Major contributions to NCAlgebra are being made by Mauricio de Oliveira and have also come from Mark Stankus and from many University of California San Diego students.