at the

American Institute of Mathematics, San Jose, California

organized by

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

- algebraic statistics,
- numerical algebraic geometry,
- toric algebraic geometry,

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.

- 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.
- 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.
- 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.

The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop:

Algorithms for Bernstein-Sato polynomials and multiplier ideals

by Christine Berkesch and Anton Leykin

Numerical algebraic geometry for Macaulay2

by Anton Leykin

Certified numerical homotopy tracking

by Carlos Beltrán and Anton Leykin