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The design and implementation of parallel algorithms is a fundamental task in computer algebra. Combining the computer algebra system Singular and the workflow management system GPI-Space, we have developed an infrastructure for massively parallel computations in commutative algebra and algebraic geometry. In this note, we give an overview on the current capabilities of this framework by looking into three sample applications: determining smoothness of algebraic varieties, computing GIT-fans in geometric invariant theory, and determining tropicalizations. These applications employ algorithmic methods originating from commutative algebra, sheaf structures on manifolds, local geometry, convex geometry, group theory, and combinatorics, illustrating the potential of the framework in further problems in computer algebra.
The Massively Parallel Computation (MPC) model serves as a common abstraction of many modern large-scale data processing frameworks, and has been receiving increasingly more attention over the past few years, especially in the context of classical gr
This is an expanded version of the two papers Interpolation of Varieties of Minimal Degree and Interpolation Problems: Del Pezzo Surfaces. It is well known that one can find a rational normal curve in $mathbb P^n$ through $n+3$ general points. More r
We give scheme-theoretic descriptions of the category of fibre functors on the categories of sheaves associated to the Zariski, Nisnevich, etale, rh, cdh, ldh, eh, qfh, and h topologies on the category of separated schemes of finite type over a separ
Particle-in-cell methods couple mesh-based methods for the solution of continuum mechanics problems, with the ability to advect and evolve particles. They have a long history and many applications in scientific computing. However, they have most ofte
We develop the framework for augmented homotopical algebraic geometry. This is an extension of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. To do so, we define the notion of augmentation categori