Macaulay2 is a free computer algebra system dedicated to the qualitative investigation of systems of polynomial equations in many variables. It was developed by Daniel Grayson and Michael Stillman with NSF funding. Grayson and Stillman will continue the development of Macaulay2. They will upgrade existing algorithms, develop and publish new algorithms, and implement new algorithms. In particular, they will develop the interaction of the symbolic computations that are Macaulay2's strength with the new floating point algorithms in algebraic geometry that are now being developed by Andrew Sommese, Jan Verschelde, Anton Leykin, Frank Schreyer and others. It is anticipated that these will make a whole new class of problems accessible to experimentation and, in many cases, solution. Eisenbud will organize contacts for the extended integration with other systems and will engage other mathematicians in the development work that needs to be done. Central to the project are the continued expansion of the collaborations that have been the hallmark of Macaulay2 development. For this purpose two Macaulay2 Workgroup Meetings will be held in the course of the two-year grant. One particular research problem to be attacked is: the use of computational systems to (probabilistically) disprove, or suggest a proof of, the Jacobian Conjecture on polynomial automorphisms of affine spaces (this will require the use the new floating point algorithms). Other 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.
Macaulay2 is part of the infrastructure that supports mathematical research involving systems of polynomial equations in many variables. The study of such systems of polynomial equations is central in pure and applied mathematics and in physics, with recent new impacts in such fields as cryptography, robotics and string theory. Increasing computer power and the availability of programs like Macaulay2 are making a new level of experimentation possible. The experimental results found with Macaulay2 are helping in the formulation and development of tractable conjectures in mathematics as well as in physics. A measure of Macaulay2's impact is that at least 270 research papers have cited Macaulay2, several mathematicians have contributed code, and books and course materials are now using it. The PI's will develop the software further and will recruit developers from the research community. They will introduce graduate students and mathematicians to the use of computers in research mathematics and the requisite skills in programming and development of algorithms, through workgroups at Berkeley and through the appointments of graduate students as graduate assistants.
