This is a collaborative project carreied out by Daniel R. Grayson and Michael E. Stillman. Macaulay 2 is a free computer algebra system dedicated to the qualitative investigation of systems of polynomial equations in many variables. The investigators continue development of Macaulay 2. They upgrade existing algorithms, install more algorithms, develop and publish new algorithms, implement new algorithms as they are discovered, add symbolic-numeric algorithm support, improve the documentation, and make systemic software changes that help contributors to write, test, and document code. Areas where new algorithms can make an impact include the study of numerical systems, fractions with specified types of denominators, ideal factorization, systems where the multiplication of the variables doesn't satisfy the commutative law, geometric optimization, the analysis of observations of gene expression levels over time, and bioinformatics. Macaulay 2 is part of the infrastructure that supports mathematical research involving systems of polynomial equations in many variables. A measure of its impact is that at least 70 research papers have cited Macaulay 2, at least seven mathematicians have contributed code, and books and course materials are now using it. The experimental results found with Macaulay 2 are helping in the formulation and development of tractable conjectures. Now that the software is approaching maturity, the investigators place more emphasis on recruiting developers from the research community and on collaborations with other scientists on both mathematical and interdisciplinary projects that exploit Macaulay 2's capabilities. They introduce graduate students and postdocs to the use of computers in research mathematics through visits and a 5 day workshop in Urbana. The study of systems of polynomial equations has a long and distinguished history in pure and applied mathematics, with recent impact on such fields as cryptography and robotics. Increasing computer power, combined with further development of Macaulay 2, enables scientists to expand dramatically its scope.
