Collaborative Research: A Software System for Algebraic Geometry Research

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The project has three main aspects. First, it addresses the infrastructure of symbolic computation as a research tool supported by Macaulay2. This includes the implementation of new features, bug fixes, user support, further documentation, and multiplatform releases of new versions that will maintain and develop Macaulay2 as a major tool used for research in a broad range of fields. Second, it addresses manpower needs, by training young people in the use of Macaulay2, and by engaging and expanding the collaborations that have been a hallmark of Macaulay2 development. In this way it will enable more of the research community to develop competence in this kind of experimental mathematics. Third, research done under this project will develop better algorithms for some key computational problems, such as computing normalization. Other goals of the research are to improve the implementation of the core Groebner basis algorithms, uncover new methods for the study of biological networks, and develop better and more reliable methods in the emerging field of numerical algebraic geometry.Macaulay2 is a free computer algebra software system dedicated to the qualitative investigation of systems of polynomial equations in many variables. The computations it can perform have uses in many fields of mathematics and science, from algebraic geometry to genomics. The research to be done under this grant extends the computational methods built into Macaulay2. In addition, the grant will provide many opportunities to train young mathematicians and other scientists in the use of these tools, and to bring other experts? knowledge to bear on them through conferences, schools, and "Intense Collaboration Workshops" centered on the important issues in this field. The project will enable new collaborations between Macaulay2 software developers from the research community and scientists from physics and biology as well as pure mathematicians. It will introduce graduate students, postdocs, and junior and senior mathematicians to the use of computers in research mathematics and help them acquire powerful skills in programming and development of algorithms that will enhance their own research. Experimental results found with Macaulay2 have helped in the formulation, development, and solution of many conjectures. Through the improvement of computational tools, the proposed research will impact many fields of mathematics and science.