##### Overview

Macaulay2 is a free open-source software for reserach in algebraic geometry and commutative algebra. Users can contribute to Macaulay2 by writing packages that extend the core functionalities.

We are organizing a Macaulay2 workshop at Cleveland State University in Cleveland, OH from Monday, May 11 to Friday, May 15, 2020. The goal of this workshop is to bring together researchers of different backgrounds and expertise levels to collaborate on developing packages for Macaulay2. We have applied for NSF funding and hope to provide support for travel and lodging (priority will go to graduate students, postdocs, and participants from underrepresented communities).

This event is organized by Federico Galetto, Courtney Gibbons, Hiram López, and Branden Stone.##### Projects

Given a reductive group acting linearly on a
polynomial ring, the ring of invariants is a
finitely generated algebra. One may wish to find
an explicit generating set for a ring of
invariants, compute its Hilbert series, or
provide bounds on the degrees of the
generators. All of this can be accomplished
computationally by means of algorithms such as
the ones described in [1,3]. The Macaulay2
package `InvariantRing` [2] implements
algorithms for finite groups. This project will
focus on implementing some of the available
algorithms for reductive groups; this includes
the general algorithm of Derksen-Kemper, as well
as more specialized algorithms for tori, Hilbert
series, and Reynolds operators of semisimple
groups.

- Harm Derksen and Gregor
Kemper.
*Computational invariant theory*. Springer, Heidelberg, enlarged edition, 2015. doi:10.1007/978-3-662-48422-7 - Thomas Hawes.
*Computing the invariant ring of a finite group*. J. Softw. Algebra Geom., 5:15–19, 2013. doi:10.2140/jsag.2013.5.15 - Bernd Sturmfels.
*Algorithms in invariant theory*. SpringerWienNewYork, Vienna, second edition, 2008. doi:10.1007/978-3-211-77417-5

Coding theory is a branch of information theory
that was originally developed to reliably
transmit information through a noisy
communication channel. Interesting parameters,
such as *dimension*
and *length*
, of
certain families of codes are related to
dimension and degree of certain algebraic
varieties, which can be studied using methods
of computational commutative algebra
(cf. [3]). Another important concept is
the *minimum distance* , which is
related to the number of errors that can be
corrected when the information is
transmitted. This notion was generalized to
graded ideals in [2], which allows to find
lower bounds for minimum distance using
Gr\"obner bases and Hilbert functions. Recent
publications (e.g. [1,4]) show Macaulay2 is
routinely used to perform computations related
to these ideas. The goal of
this project is to review the existing
literature for procedures and algorithms that
can be coded and distributed as a package to
streamline future computations in coding theory
using Macaulay2.

- Susan M. Cooper, Alexandra Seceleanu,
Ştefan O. Tohăneanu, Maria Vaz
Pinto, and Rafael H. Villarreal.
*Generalized minimum distance functions and algebraic invariants of Geramita ideals.*Adv. in Appl. Math., 112:101940, 2020. doi:10.1016/j.aam.2019.101940 - José Martínez-Bernal, Yuriko
Pitones, and Rafael
H. Villarreal.
*Minimum distance functions of graded ideals and Reed-Muller-type codes.*J. Pure Appl. Algebra, 221(2):251–275, 2017. doi:10.1016/j.jpaa.2016.06.006 - Carlos Rentería-Márquez,
Aron Simis, and Rafael
H. Villarreal.
*Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields*. Finite Fields Appl., 17(1):81–104, 2011. doi:10.1016/j.ffa.2010.09.007 - Manuel Gonzalez Sarabia, Miguel
E. Uribe-Paczka, Eliseo Sarmiento, and
Carlos Renteria.
*Relative generalized minimum distance function.*2019. arXiv:1907.11324

The current `SymbolicPowers` package
consists of various tools for computations
related to symbolic powers of ideals. This is
an ubiquitous subject within commutative
algebra, with connections to algebraic
geometry, combinatorics, and arithmetic
geometry, among other fields. There are
various open problems related to symbolic
powers that are the subjects of very active
current research, including the Containment
Problem, the Uniform Symbolic Topologies
problem, equality of symbolic and ordinary
powers, in particular with questions such as
the Packing Problem, or noetherianity of
symbolic Rees algebras [1]. One of the
obstacles to solving these problems is the
fact that actually computing symbolic powers
is fairly difficult, and this is an issue
the `SymbolicPowers` package aims to
improve. While symbolic powers can be computed
via primary decompositions of the
corresponding powers, computing primary
decompositions is notably slow, and
the `SymbolicPowers` package provides
functionality that bypasses the computation of
explicit primary decompositions. However, the
package's current functionality is mostly
directed at radical ideals in regular
rings. During the workshop, participants will
focus on the development of new algorithms for
computing symbolic powers based on the current
literature.

- Hailong Dao, Alessandro De Stefani,
Eloísa Grifo, Craig Huneke, and
Luis
Núñez-Betancourt.
*Symbolic powers of ideals.*Springer Proceedings in Mathematics & Statistics. Springer, 2017. doi:10.1007/978-3-319-73639-6_13

The goal of this package is to develop
functionality for using the simplex method and
other techniques for solving combinatorial
optimization problems. Such problems arise
frequently for many researchers who use
Macaulay2 (in toric geometry or
Boij-Söderberg theory [1], for example),
and folding these functions into Macaulay2
would allow them to be more easily used by
researchers. This package will also introduce
functionality for solving Linear Programming
problems in characteristic
*p*.

- David Eisenbud and Frank-Olaf
Schreyer.
*Betti numbers of graded modules and cohomology of vector bundles.*J. Amer. Math. Soc., 22(3):859–888, 2009. doi:10.1090/S0894-0347-08-00620-6

Currently there is a single package dealing
with statistics in
Macaulay2, `Markov`. This package
constructs Markov ideals, arising from
Bayesian networks in Statistics. Due to the
development of Algebraic Statistics, a new
package would be useful in order to
consolidate and extend methods to calculate
Maximum Likelihood Estimate degree for
specific ideals and varieties; enhance and
refine homotopy continuation methods for
different parametrizations of statistical
models [1,2].

- Carlos Améndola, Nathan Bliss,
Isaac Burke, Courtney R. Gibbons, Martin
Helmer, Serkan Hoşten, Evan D. Nash,
Jose Israel Rodriguez, and Daniel
Smolkin.
*The maximum likelihood degree of toric varieties.*J. Symbolic Comput., 92:222–242, 2019. doi:10.1016/j.jsc.2018.04.016 - Jose Israel Rodriguez.
*Solving the likelihood equations to compute Euler obstruction functions.*Springer, Cham, 2018. doi:10.1007/978-3-319-96418-8_48

##### Schedule

##### Confirmed participants

David Eisenbud — University of California, Berkeley |

Daniel Grayson — University of Illinois at Urbana-Champaign |

Gregory Smith — Queen’s University |

Michael Stillman — Cornell University |