Overview

We are organizing a Macaulay2 conference at Cleveland State University in Cleveland, OH from Friday, May 27 to Sunday, May 29, 2022 (conference activities running from Friday afternoon through Sunday morning).

The main goal of the conference is to showcase state-of-the-art research in computational commutative algebra, algebraic geometry, and related areas, including (but not limited to) work resulting from the 2020 M2@CSU workshop. In addition, we hope the conference will provide an ideal forum for the exchange of mathematical ideas in our community. In order to facilitate social interactions among colleagues, we will focus primarily on in-person participation; however, we also plan to stream talks on Zoom (please contact the organizers for the link).

For planning purposes, we kindly request all interested participants to register here. NSF funding will provide partial support for travel and lodging (priority will go to graduate students, postdocs, and participants from underrepresented communities). If you wish to apply for this kind of support, we ask that you submit your application by January 31, 2022.
Internals meeting
The main M2internals activities will take place in the afternoon of May 26th and the morning of May 27th. If interested in joining, please look at the notes from previous meetings to see what this is about. It may also help to see examples of engine projects.
Code of Conduct
  • All visitors to campus must adhere to applicable CSU policies. Notice in particular the CSU policy against discrimination, harassment, sexual violence, and retaliation.
  • We are dedicated to providing a harassment-free conference experience for everyone, regardless of gender, gender identity and expression, age, sexual orientation, disability, physical appearance, race, ethnicity, national origin, or religion (or lack thereof). We do not tolerate harassment of conference participants in any form. Any incidents may be reported to the organizers or CSU police.
  • Effective May 21, 2022, masking on the CSU campus is optional, and there are no social distancing requirements. Conference participants are welcome to wear masks, and we ask that all attendees respect the choices of other people on campus. The CDC recommends that COVID-positive individuals isolate at home for 5 days after symptoms develop, and wear a mask for an additional 5 days when returning to public places. Please check the CSU COVID-19 safety protocols for updates before coming to campus.
Venue and Local Information
  • Conference events will take place in Berkman Hall 445 on the Cleveland State University campus at 2121 Euclid Avenue, Cleveland, OH. Accessible entrance and elevators are located at the corner of East 22nd Street and Chester Avenue.
  • The Internals workshop will take place in Rhodes Tower 1402. Accessible entrance to Rhodes Tower is available from Euclid Avenue (around or through the Student Center, then across the plaza), and from the garage level on East 21st or East 22nd Street.
  • Here is a campus map and a map of parking locations.
  • Here is some accessibility information for visitors to the CSU campus.
  • Inclusive multi-stall restrooms are located in front of the conference room, and more restrooms are available on the same floor closer to the center of the building. Here is a list of gender neutral restrooms on campus.
  • Internet access for campus visitors is provided by Eduroam or by the (unsecured) CSUguest network.
  • The map below shows some restaurants and coffee shops near the CSU campus. View it on Google Maps.
Speakers
(click name to expand)
Subalgebra bases are an analogue of Groebner bases for polynomial algebras. In this talk, I will introduce the newly revised SubalgebraBases package. This package contains routines for computing subalgebra bases as well as tools for applying these bases.
Matroids are combinatorial structures that arise naturally in diverse areas of mathematics. The simplest and most ubiquitous class of matroids are representable matroids (over a field), which are fully captured by a finite set of vectors. However, finding a representation for a given matroid is a difficult problem. I will discuss how recent work by Baker-Lorscheid enables efficient computation of representations of matroids over finite fields, using algebraic structures called pastures (which generalize fields). This is joint work with Tianyi Zhang.
Error correction in digital information is crucial in modern communication devices. Recently, a Coding Theory package has been developed for Macaulay2. Although many functions and constructions of families of error correcting codes are included there, many others are missing. Instead of presenting work that has been done in Coding theory with the Help of Macaulay2, In this talk we will propose the implementation of more functions to complement the Coding Theory package of Macaulay2. First, we will propose to implement functions to describe the algebraic structure of Quasi-Cyclic codes. Quasi-cyclic codes are among the most important families of codes because they attain the highest minimum Hamming distance possible for a given length and dimension. On the other hand, we will propose to implement functions to study codes over finite rings. These are of interest to the research community in Coding theory as they close relation to other areas in Discrete Mathematics and DNA coding.

Abelian groups -- modules over the integers -- are well-described by giving generators and relations, conveniently described by one matrix of integers. For modules over a polynomial ring in n variables David Hilbert improved the description to one requiring n matrices, the "free resolution". But for most commutative rings, the corresponding description requires infinitely many matrices.

In the finite case, we know a lot, and in the first half of this talk I will survey some of what we've learned in the 130 years since Hilbert's work. But except for very special cases, we know almost nothing in the infinite case. In the second half of the talk I will introduce some new problems and conjectures that Hai Long Dao and I have been working on, with the help of the program Macaulay2.
We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.
We present the package MixedMultiplicity for computing mixed multiplicities of ideals in a Noetherian ring. This enables us to find mixed volumes of convex lattice polytopes and sectional Milnor numbers of hypersurfaces with an isolated singularity. The algorithms require the computation of the defining equations of the multi-Rees algebra of ideals, a task we accomplish using a generalization of a result of David A. Cox, Kuei-Nuan Lin, and Gabriel Sosa.
We give a definition of the s-order jets of an ideal of a polynomial ring and show how the Jets package of Macaulay2 can be used to calculate the jets of an ideal. We then show how we can use the jets of an ideal to calculate the jets of some other algebraic objects, including edge ideals of graphs. Finally, we define the s-order principal component of the edge ideal of a graph and present an application to the cochordal family of graphs.
In this talk, we will consider the edge ideals of weighted oriented graphs. In the first half of the talk, we will survey some of the results about the algebraic invariants of these monomial ideals. The second half of the talk will focus on Morse resolutions of monomial ideals and we will discuss when these resolutions are minimal. As an application, we will present minimal cellular free resolutions of edge ideals of certain classes of weighted oriented graphs.
We present the Macaulay2 package NumericalCertification for certifying the correctness of approximations for roots of square polynomial systems. It employs the Krawczyk method and α-theory as the main methods for certification. The package works with output data computed in Macaulay2 without using external software. Also, we introduce potential extensions of the package to the multiple roots certification.
The task of describing the complex variety cut out by a sequence of polynomial equations is one of the central tasks of numerical algebraic geometry. We develop a new geometric construction that addresses this "equation by equation". The resulting algorithm called "u-generation" uses a smaller number of continuation paths in comparison to the state-of-the-art analogs. Apart from theoretical advances, I will demonstrate Macaulay2 tools necessary for an implementation of u-generation. (This is joint work with Tim Duff and Jose Rodriguez.)
Computing the minimum distance of a linear code is an NP-hard problem that is relevant to certain mathematical research and engineering applications. In this talk, I will explain reasons why computing the minimum weight of a linear code is an important feature in a software package for coding theory research. I will also give an overview of the state of our minimum weight algorithm implementation in the package CodingTheory, including how it compares with other available implementations as well as possibilities for improvement.
The Macaulay2 package AlgebraicOptimization implements methods for determining the algebraic degree of an optimization problem. In this talk I will describe the structure of an algebraic optimization problem and explain how the methods in this package may be used to determine the respective degrees. Special features include determining Euclidean distance degrees and maximum likelihood degrees. The package is available at https://github.com/Macaulay2/Workshop-2020-Cleveland/tree/ISSAC-AlgOpt/alg-stat/AlgebraicOptimization. Time permitting, I will also how this is related to other M2 packages.
I will talk about how to use the NormalToricVarieties package, focusing first on defining toric varieties from combinatorial data, the Cox ring, and working with divisors and sheaves, and second on features such as toric maps and Chern classes.
We will examine a few open problems and speculate about why this might be a fruitful area for computational experimentation. We will also showcase some new calculations in the developing ToricReflexiveSheaves package.
What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this talk, I invoke the Bernstein--Gelfand--Gelfand (BGG) correspondence to address this question for free resolutions given by two well-known complexes, the Koszul and the Eagon--Northcott. This novel approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon--Northcott case. This is joint work with Maya Banks.
Schedule
Photos