Implementing algebraic geometry algorithms

October 26 to October 30, 2009

at the

American Institute of Mathematics, Palo Alto, California

organized by

Hirotachi Abo, Anton Leykin, Sam Payne, and Amelia Taylor

Original Announcement

This workshop will be devoted to developing three packages,
  1. algebraic statistics,
  2. numerical algebraic geometry,
  3. toric algebraic geometry,
for the computer algebra system Macaulay 2. Macaulay 2 is a widely used computer algebra system for research and teaching in algebraic geometry and commutative algebra and is one of the leading computer algebra programs for performing such computations.

These three topics are all very active areas of research in computational algebra and algebraic geometry and are linked in surprising ways which lends them nicely to be the three packages of focus for this workshop.

  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.

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:
Algorithms for Bernstein-Sato polynomials and multiplier ideals
Numerical algebraic geometry for Macaulay2
Certified numerical homotopy tracking