

ICMS 2024  Session 5 

Advancing computer
algebra with massively parallel methods 



Organizers 












Aim and Scope 








The session will highlight how these massively parallel methods are transforming various areas of research in computer algebra. The core of our session will be the innovative use of massively parallel methods across different areas, with a specific focus on computational algebraic geometry and applications. In high energy physics, massively parallel methods are reshaping the way we approach multiloop Feynman integral reduction in perturbative quantum field theory. Another focus will be on integrable systems, particularly regarding spin chains and the Bethe Ansatz equation. We also plan to touch the computation of generating series for GromovWitten invariants, as well as parallel algorithmic methods for algebraic schemes. The Computeralgebra/GPISpace Framework, as an example of advanced software enabling massively parallel methods, will be mentioned in the context of its integration with computer algebra systems like Singular and OSCAR. The primary emphasis will be on the broader application and implications of massively parallel methods. The session would aim at a series of talks offering a distinct perspective on the current state of the art and future potential in computer algebra and applications. The organizers will give an introduction, setting the stage for a series of invited presentations. 






Schedule 










Talks 






Overview Talk: Parallel Methods in Commutative Algebra and Algebraic Geometry: Past Successes and Future Challenges 


Janko Boehm (RPTU, Germany), Anne FruehbisKrueger (University of Oldenburg, Germany) 



Abstract: The introduction of massively parallel methods in commutative algebra and algebraic geometry is offering a game changing improvement of efficiency and scalability in experimental techniques as well as applied computations. This talk aims to provide an overview of the potential of parallel methods within related fields. We will look at notable examples where parallel methods have successfully been applied, and give ideas of current developments in the areas of Gröbner bases, syzygies, modular methods, birational geometry, and high energy physics. The presentation will also address ongoing efforts to establish a generic programming model, as well as potential future applications. 







Using algebraic geometry to reduce Feynman integrals: automatic software NeatIBP 


Yang Zhang (University of Science and Technology of China, Hefei, Anhui, China) 



Abstract: We present the package NeatIBP, which automatically generates smallsize integrationbyparts (IBP) identities for Feynman integrals. Based on the syzygy and module intersection techniques, the generated IBP identities' propagator degree is controlled and thus the size of the system of IBP identities is shorter than that generated by the standard Laporta algorithm. The computation powered by Singular and Mathematica. The parallelization is implemented on the level of subdiagrams of Feynman integral via a managerworker model. The future development will focus on the fully opensource implement and more layers of syzygy computations and Feynman integral relation searching. 







Massively Parallel Modular Algorithms for the Image of Rational Maps 



Abstract: Computations over the rational numbers frequently encounter the issue of intermediate coefficient growth. Modular methods provide a solution to this problem by applying the algorithm under consideration modulo a number of primes and then lifting the modular results to the rationals. We focus on the application of these methods within commutative algebra and algebraic geometry, showcasing their implementation in a massively parallel framework in Groebner basis computations and various algorithmic strategies in birational geometry. Specifically, we will address algorithms for computing the images and domains of rational maps, assessing invertibility, and deriving inverses. The effectiveness of our modular approache is evidenced through timings, which demonstrate significant performance enhancements over traditional modular and nonmodular methods. 







Generic modular methods for commutatve algebra and algebraic geometry 



Abstract: This presentation discusses a massively parallel framework for modular computations with polynomial data, leveraging the capabilities of the Singular/GPISpace framework. Singular serves as the computational backbone, providing the algebraic engine, while GPISpace manages the orchestration and communication of parallel tasks, relying on Petri nets as a mathematical modeling language. Convenient installation is realized through the Spack package manager. Our framework's strength lies in automated parallelization and balancing of the load between computation, lifting, stabilization testing, and potential verification. We use error tolerant rational reconstruction to ensure termination as long as for a fixed computation there exist only finitely many bad primes. Via stabilization testing, our approach automatically finds with high probablity a minimal set of primes required for the successful reconstruction. We will also discuss current work towards establishing a generic programming model that can further democratize access to massively parallel methods for commutative algebra and algebraic geometry, both for mathematians as well as applied scientists. 







Massively parallel methods in birational geometry 

Benjamin Mirgain (Universität des Saarlandes, Fraunhofer ITWM) 

Abstract: Computational birational geometry is one of the key playing fields in an algorithmic approach to algebraic geometry, since birational maps are the fundamental way to relate algebraic varieties (or schemes). An important application is an algorithmic approach to the Minimal Model Program (MMP), which aims to classify algebraic varieties with mild singularities by finding simple birational models of such varieties in their birational equivalence class. The aim of this talk is to develop massively parallel methods to solve problems in birational geometry and the MMP. Making use of a representation of algebraic varieties in terms of charts and employing modular methods for problems in characteristic zero, allows for a massively parallel computational approach for handling both the varieties and rational maps between them. For the MMP we aim to implement and utilize a proof of the finite generation of the cannonical ring as developed by LaziÄ‡. 







Algorithms for GromovWitten invariants of elliptic curves 



Abstract: We present an enhanced algorithm for exploring mirror symmetry for elliptic curves through the correspondence of algebraic and tropical geometry, focusing on GromovWitten invariants of elliptic curves and, in particular, Hurwitz numbers. We present a new highly efficient algorithm for computing generating series for these numbers. We have implemented the algorithm both using Singular and OSCAR. The implementations outperform by far the current method provided in Singular. The OSCAR implementation, benefiting in particular from justintime compilation, again by far outperforms the implementation of the new algorithm in Singular. This advancement in computing the GromovWitten invariants facilitates a study of number theoretic and geometric properties of the generating series, including quasimodularity and homogeneity. 







TBA 

TBA (Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Zhejiang, China) 



Abstract: TBA 







Massively Parallel Methods for Free Resolutions 







Abstract: This talk introduces an approach aimed at applying massively parallel methods for computing syzygies and free resolutions, with a primary focus on Schreyer's Resolution. Our method capitalizes on the inherent parallelism of the algorithm, primarily utilizing Petri nets, within the GPISpace framework, as our key language for parallelism. GPISpace is a taskbased workflow management system that employs Petri nets as its coordination layer, while the computation is carried out by the computer algebra system Singular We present in detail the way the algorithm is modelled through a Petri net, explaining the coordination of tasks and data structures within the parallel computing environment. Although our implementation is still workinprogress, preliminary results show promising advancements in efficiency. By exploiting the potential synergy between the computer algebra system Singular and the GPISpace, we anticipate significant reductions in computational time for computing free resolutions. 








Santiago Laplagne (Instituto de Cálculo  FCEyN  UBA, Buenos Aires) 

TBA 



Abstract: TBA 





