Collaborative Research: A Software System for Research in Algebraic Geometry, Commutative Algebra, and their Applications

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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. 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. 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.The project has two 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. Secondly, research done under this project will develop better algorithms for some key computational problems, such as computing the cohomological and topological invariants of algebraic varieties, and will develop applications to string theory, supersymmetry, and discrete models of biological networks.