Implementing algebraic geometry algorithms
October 26 to October 30, 2009
American Institute of Mathematics,
Palo Alto, California
and Amelia Taylor
This workshop will be devoted to developing
for the computer algebra system Macaulay 2. Macaulay 2 is a widely
used computer algebra system for research and teaching in algebraic
commutative algebra and is one of the leading computer algebra programs
performing such computations.
- algebraic statistics,
- numerical algebraic geometry,
- toric algebraic geometry,
These three topics are all very active areas of research in computational
and algebraic geometry and are linked in surprising ways which lends them
to be the three packages of focus for this workshop.
- Algebraic Statistics: Some of the key varieties arising in the
of algebra and geometry to phylogenetics are toric, while other challenges
studying both phylogenetics and reverse engineering of biochemical systems
rooted in the need for better numerical techniques for algebraic geometry.
is also the case that solving such problems, and related problems more
in algebraic statistics, often require non-standard approaches to
primary decompositions and other standard algebraic objects for which
available code might allow form greater experimentation and study.
- Numerical Algebraic Geometry: While there are tasks best accomplished
numerically and other tasks that can be approached only symbolically,
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.
- Toric Geometry: Toric geometry stands at the interface between
algebra, combinatorics, and geometry and has a rich history as a testing
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
normalized Nash blowups resolve arbitrary singularities and computing
sets of examples of normalized Nash blowups of higher dimensional toric
varieties, computing weighted Ehrhart series, and implementation of
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: