A Software System for Algebraic Geometry Research

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Stillman9970348 Stillman and his colleague Grayson collaborate in developingthe computer software system Macaulay 2, a symbolic computeralgebra program that supports large scale research computingprojects dealing with systems of polynomial equations in manyvariables. The program's many functions depend mainly onvariants of the Groebner basis algorithm for computing standardbases and syzygies of ideals and modules over polynomial rings.The results yield quantitative and qualitative information aboutsolutions of such systems of equations. In this project theinvestigators develop new algorithms, improve existing algorithmsand implementations, improve the documentation, add new features,improve the program's efficiency, and make it more useable. Theyalso provide a web-accessible computation server to facilitatecontributions from and collaborations between the Internet-basedcommunity of users. These interchanges improve the algorithmsand scope of Macaulay 2. Stillman also continues his research incomputational commutative algebra and algebraic geometry, as wellas into problems involving generalized Green's conjectures,Castelnuovo-Mumford regularity, toric varieties and binomialideals. Grayson also continues his research into algebraicK-theory concerning the connection between algebraic K-theory andmotivic cohomology as mediated by filtrations of the K-theoryspace. The goal of this project is to turn Macaulay 2 into thepremiere collaborative research tool for mathematicians andothers around the world whose work involves systems of polynomialequations. Such research is important because systems ofpolynomial equations arise whenever mathematics is used as a toolin science, for example, in engineering, in physics, and inbiology. The project extends the algorithms, efficiency, andscope of Macaulay 2, provides an Internet server for users, andprovides an Internet-based forum for users to contribute Macaulay2 packages and to collaborate on the solution of mathematics andapplications problems. The project also involves the training ofstudents in both mathematics and symbolic computation.