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.

• Video

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.

• Video
• Slides

• Video

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.

• Video
• Slides

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.

• Video
• Slides
• OSCAR System

Abstract

Introducing OSCAR, a new computer algebra system combining GAP, Polymake, Hecke and Singular.

• Slides

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.

• Video
• Slides

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.

• Video

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.

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.

• Video

• Slides
• Abstracts

• Slides
• Abstracts

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.

• Video

Abstract

I shall illustrate the use of the Singular - GPI-space interplay in some examples including a smoothness test, GIT-fans and desingularization.

• Slides

• Video

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.

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.

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.

• Video

• Video