Organizing Committee
Abstract

This workshop will focus on the development of software to facilitate research in combinatorial algebraic geometry, based on the SAGE Mathematical Software System and the OSCAR Computer Algebra System. Special attention will be given to efficient computations with multi-variate polynomials, which is a critical part of many algorithms in combinatorial algebraic geometry. Aside from development of software, the workshop will feature a series of talks about polynomial computations, as well as introductory lectures about Sage and Oscar.

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Confirmed Speakers & Participants

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

Monday, February 15, 2021
  • 9:45 - 10:00 am EST
    Welcome
    Virtual
    • Brendan Hassett, ICERM/Brown University
  • 10:00 - 10:30 am EST
    A brief tour of Sage
    Virtual
    • Nicolas Thiéry, Université Paris Sud
    Abstract
    I will offer a brief tour of Sage, showcasing some features and use cases, hinting at its development model, pointing to some recent trends, and highlighting how it fits within the larger ecosystem of free computational (mathematics) software.
  • 10:45 - 11:15 am EST
    Rings and fields in Sage
    Virtual
    • David Roe, Massachusetts Institute of Technology
    Abstract
    I will give an introduction to basic algebraic structures in Sage, with a focus on the coercion model, finite fields and extensions of rings. I will also give an overview of how you can contribute to Sage.
  • 11:15 - 11:30 am EST
    Coffee Break
    Virtual
  • 11:30 am - 12:00 pm EST
    Celestial mechanics via tropical geometry (gfan and Macaulay2)
    Virtual
    • Anton Leykin, Georgia Tech
  • 12:15 - 12:45 pm EST
    FusionRings in Sage 9.2
    Virtual
    • Daniel Bump, Stanford University
    Abstract
    The FusionRing class implements useful methods for Verlinde Algebras. These are elegant rings similar to WeylCharacterRings (representation rings of Lie groups) except that the fusion categories have only finitely many objects. These rings have applications to conformal field theory, quantum groups, topological quantum computing and knot theory. Most of the methods needed to work with these have been implemented in Sage 9.2. We will review the math and show what the code can do. The FusionRing code is joint work with Guillermo Aboumrad.
  • 1:00 - 2:00 pm EST
    Lunch/Free Time
    Virtual
  • 2:00 - 3:00 pm EST
    Gathertown Welcome Reception
    Reception - Virtual
  • 3:00 - 4:00 pm EST
    Sage/Oscar Installation Help
    Tutorial - Virtual
Tuesday, February 16, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:30 am EST
    OSCAR - The Project
    Virtual
    • Michael Joswig, TU Berlin & MPI Leipzig
    Abstract
    The OSCAR project is a collaborative effort to shape a new computer algebra system, written in Julia. OSCAR is built on top of the four "cornerstone systems" ANTIC (for number theory), GAP (for group and representation theory), polymake (for polyhedral and tropical geometry) and Singular (for commutative algebra and algebraic geometry). We present three examples to showcase the current version 0.5.1. This is joint work with The OSCAR Development Team.
  • 10:45 - 11:15 am EST
    OSCAR - Selected Features
    Virtual
    • Daniel Schultz, Technische Universität Kaiserslautern
    Abstract
    Introducing OSCAR, a new computer algebra system combining GAP, Polymake, Hecke and Singular.
  • 11:15 - 11:30 am EST
    Coffee Break
    Virtual
  • 11:30 am - 12:00 pm EST
    Computing the Newton polytope of a large discriminant
    Virtual
    • Lars Kastner, Institute of Mathematics of the Technical University
    Abstract
    The Newton polytope of the discriminant of a quaternary cubic form has 166'104 vertices. One way to obtain these vertices is to enumerate all D-equivalence classes of regular triangulations of the 3- dilated tetrahedron. The only known way to do this is to enumerate all regular triangulations of the 3-dilated tetrahedron and group them into classes in a second step. This talk will focus on the computations carried out to arrive at this result. It involved the use of polymake and mptopcom on large computing clusters in parallel which in turn brought other obstacles. This software can also be used via polymake.jl in OSCAR. Since computer experiments in algebraic geometry are becoming larger and larger, this talks aims at providing insights on how to set up these experiments such that they give reliable results, and how to avoid the pitfalls we encountered. This is joint work with Robert Loewe.
  • 12:15 - 12:45 pm EST
    Some hybrid symbolic-numeric methods in algebraic geometry
    Virtual
    • Jonathan Hauenstein, University of Notre Dame
    Abstract
    On the theoretical side, algebraic geometry combines aspects of algebra and geometry to provide many tools to prove new results. On the computational side, symbolic computations typically based on algebra and numerical computations typically based on geometry can be combined to provide many new computational tools to study a variety of problems in algebraic geometry. This talk will explore some hybrid symbolic-numeric methods and applications in computational algebraic geometry.
  • 1:00 - 2:00 pm EST
    Lunch/Free Time
    Virtual
  • 2:00 - 3:00 pm EST
    Problem Session
    Virtual
  • 3:00 - 4:00 pm EST
    Contributing to Sage Tutorial
    Tutorial - Virtual
Wednesday, February 17, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EST
    Parallelization of Triangular Decompositions- Design and implementation with the BPAS library
    Virtual
    • Marc Moreno, University of Western Ontario
    Abstract
    We discuss the parallelization of algorithms for solving polynomial systems by way of triangular decomposition. The "Triangularize" algorithm proceeds through incremental intersections of polynomials to produce the different components of the solution set. Independent components imply the opportunity for concurrency. This "component-level" parallelization of triangular decompositions, our focus here, belongs to the class of dynamic irregular parallelism. Potential parallel speed-up depends only on geometrical properties of the solution set (number of components, their dimensions and degrees); these algorithms do not scale with the number of processors. To manage the irregularities of component-level parallelization we combine different concurrency patterns: map, workpile, producer-consumer, pipeline and fork/join. We report on our implementation in the freely available BPAS library. Comprehensive experimentation with thousands of polynomial systems yields examples with up to 10.8-times speed up on a 12-core machine.
  • 11:00 - 11:30 am EST
    Coffee Break
    Virtual
  • 11:30 am - 12:00 pm EST
    Rational integrals and periods with Sagemath and Julia
    Virtual
    • Pierre Lairez, INRIA
    Abstract
    Based on symbolic integration and numerical analytic continuation, we can compute to high precision integrals of multivariate rational functions. I will show applications to volume computation and to the study of quartic surfaces. I will emphasize on some software aspects, specific to Sagemath and Julia.
  • 12:15 - 12:45 pm EST
    Generalized cohomology quotients of the symmetric functions
    Virtual
    • Darij Grinberg, Drexel University
  • 1:00 - 2:00 pm EST
    Lunch/Free Time
    Virtual
  • 2:00 - 2:40 pm EST
    Lightning Talks
    Virtual
    • Adam Afandi, Colorado State University
    • Jose Brox, Centre for Mathematics of the University of Coimbra
    • Juliette Bruce, University of California, Berkeley / MSRI
    • Laura Brustenga i Moncusi, University of Copenhagen
    • Taylor Brysiewicz, Max Planck Institute for Mathematics in the Sciences
    • Papri Dey, University of Missouri
    • Sean Griffin, Brown University
    • Shinyoung KIM, Institute for Basic Science Center for Geometry and Physics
  • 2:40 - 2:50 pm EST
    Coffee Break
    Virtual
  • 2:50 - 3:30 pm EST
    Lightning Talks
    Virtual
    • Lukas Kühne, Max Planck Institute for Mathematics in the Sciences
    • Jianping Pan, University of California, Davis
    • Marta Panizzut, TU Berlin
    • Theodoros Stylianos Papazachariou, University of Essex
    • Colleen Robichaux, University of Illinois at Urbana-Champaign
    • Mahrud Sayrafi, University of Minnesota
    • Weihong Xu, Rutgers
  • 3:30 - 4:30 pm EST
    Code Demonstrations
    Tutorial - Virtual
Thursday, February 18, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:45 am EST
    msolve - A Library for Solving Polynomial Systems
    Virtual
    • Christian Eder, University of Kaiserslautern
    Abstract
    We present a new open source C library msolve dedicated to solve multivariate polynomial systems exactly through computer algebra methods. The core algorithmic framework of msolve relies on Gröbner bases and linear algebra based algorithms for polynomial system solving. It relies on Gröbner basis computation w.r.t. the degree reverse lexicographical order, Gröbner conversion to a lexicographical Gröbner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented exploiting the computational capabilities of the framework. We compare the practical performance of the different parts of msolve with similar functionalities of leading computer algebra systems such as Magma and Maple on a wide range of polynomial systems with a particular focus on those which have finitely many complex solutions, showing that msolve can tackle systems which were out of reach by the software state-of-the-art. This is joint work with Jérémy Berthomieu, Jean-Charles Faugère and Mohab Safey El Din from the PolSys Team at the Sorbonne Université in Paris.
  • 11:00 - 11:30 am EST
    Parallelism in Algebraic Geometry - Examples with Singular and GPI-Space
    Virtual
    • Anne Frühbis-Krüger, University of Oldenburg
    Abstract
    I shall illustrate the use of the Singular - GPI-space interplay in some examples including a smoothness test, GIT-fans and desingularization.
  • 11:45 am - 12:45 pm EST
    Coffee Break
    Virtual
  • 12:45 - 1:15 pm EST
    Computing Galois groups in enumerative geometry
    Virtual
    • Frank Sottile, Texas A&M University
  • 1:30 - 2:30 pm EST
    Lunch/Free Time
    Virtual
  • 2:30 - 3:30 pm EST
    Problem Session
    Virtual
  • 3:30 - 4:30 pm EST
    Code Demonstrations
    Tutorial - Virtual
Friday, February 19, 2021
  • 9:00 - 9:45 am EST
    Gathertown Morning Coffee
    Coffee Break - Virtual
  • 10:00 - 10:30 am EST
    Factorizations into irreducibles and polytopes
    Virtual
    • Tommy Hofmann, TU Kaiserslautern
    Abstract
    Dedekind domains form a family of commutative rings that plays an important role in algebraic geometry and number theory. While elements of Dedekind domains factor into irreducible elements, such a factorization is in general not unique. We present an algorithm, which for a given element of the ring of integers of a number field, determines all factorizations into irreducible elements. The algorithm makes heavy use of computations with polytopes and is implemented in Oscar. This is joint work with Claus Fieker.
  • 10:45 - 11:30 am EST
    Computational challenges for tropical del Pezzo surfaces
    Virtual
    • María Angélica Cueto, Ohio State University
    Abstract
    A smooth degree d del Pezzo surface is obtained by blowing up the projective plane at (9-d) generic points. In this talk, we will discuss how to tropicalize these surfaces for various embeddings as we vary the input points and the computational challenges that arise when doing so.
  • 11:15 - 11:30 am EST
    Coffee Break
    Virtual
  • 11:30 am - 12:00 pm EST
    Presenting the multipolynomial bases package
    Virtual
    • Viviane Pons, Université Paris Sud
    Abstract
    In this talk, we present an external SageMath package to work on multivariate polynomials seen as an algebra over integer vectors (the exponents). This allows for manipulation of divided differences operators and the definition of many bases of multivariate polynomials such as the Schubert polynomials, Grothendieck, and Demazure Characters.
  • 12:15 - 12:45 pm EST
    Hasse-Witt matrices and mirror toric pencils
    Virtual
    • Adriana Salerno, Bates College
  • 1:00 - 2:00 pm EST
    Lunch/Free Time
    Virtual
  • 2:00 - 3:00 pm EST
    Gathertown Closing Reception
    Reception - Virtual

All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC-5).

All event times are listed in .

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Associated Semester Workshops

Combinatorial Algebraic Geometry
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Schubert Seminar Series
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VIRTUAL ONLY: Algebraic Geometry and Polyhedra
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