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
Please let us know about projects you are interested in when you register for the workshop. The following are sample projects. For more information about packages under development please visit https://faculty.math.illinois.edu/Macaulay2/dev/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.

  1. Harm Derksen and Gregor Kemper. Computational invariant theory. Springer, Heidelberg, enlarged edition, 2015. doi:10.1007/978-3-662-48422-7
  2. 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
  3. 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.

  1. 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
  2. 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
  3. 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
  4. 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.

  1. 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.

  1. 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].

  1. 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
  2. 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