Implementing Algebraic Geometry Algorithms

This conference will be focused on three topics:

1. algebraic statistics,
2. numerical algebraic geometry,
3. toric algebraic geometry,

1. Algebraic Statistics: Some of the key varieties arising in the application of algebra and geometry to phylogenetics are toric, while other challenges in studying both phylogenetics and reverse engineering of biochemical systems are rooted in the need for better numerical techniques for algebraic geometry. It is also the case that solving such problems, and related problems more broadly in algebraic statistics, often require non-standard approaches to computing primary decompositions and other standard algebraic objects for which broadly available code might allow form greater experimentation and study.

    

2. Numerical Algebraic Geometry: While there are tasks best accomplished numerically and other tasks that can be approached only symbolically, there is a multitude of problems in computational algebraic geometry currently unsolved by either. A system which allows a user to seamlessly access both the numerical and symbolic algorithms and to write hybrid programs will make possible the kind of experimentation that might solve these problems. Developing the ability to create hybrid programs is the primary focus of this package. Developing such a package requires a combination of a clear understanding of both numerical methods and current problems in algebra and geometry that might benefit from this package, like algebraic statistics and toric algebraic geometry.

    

3. Toric Geometry: Toric geometry stands at the interface between commutative algebra, combinatorics, and geometry and has a rich history as a testing ground for emerging theories and general conjectures in algebraic geometry. Several topics of current research are suitable for computational exploration, and access to efficient software could lead to rapid and significant progress on open problems, including determining whether iterated normalized Nash blowups resolve arbitrary singularities and computing large sets of examples of normalized Nash blowups of higher dimensional toric varieties, computing weighted Ehrhart series, and implementation of algorithms in toric intersection theory.