Recent Trends in Computer Algebra

Du 1 avril 2023 au 15 juillet 2023

Lyon / Paris 

Computer algebra, a.k.a. symbolic computation, is a broad transdisciplinary area which aims at computerizing mathematics, i.e. solving exactly mathematical problems, using computers. Hence, it encompasses effective mathematics in algebra, analysis, geometry and number theory, the design of algorithms, the study of their complexities, their implementations and their use in applications, as well as software system aspects to manipulate and encode efficiently mathematical objects.

Because of the sophistication of these high performance implementations, combined with the use of Monte Carlo probabilistic algorithms, reliability issues were raised. Certification from various viewpoints and methodologies emerged.

Especially, interactions between computer algebra and theorem provers increased significantly during the last decade. In parallel, the design and computation of certificates that can be efficiently checked a posteriori arose as a fundamental problem from the angles of algebraic complexity, interactive proof and the numerical hybridization of some computer algebra routines. These active trends are nowadays considered crucial to the development of reliable experimental mathematics through computer algebra.

Still, besides the progress made these last years, many important industrial and theoretical applications such as polynomial optimization problems (for energy minimization), analysis of integrable models (in theoretical physics) or topological analysis of semi-algebraic sets (for mechanism design) require further developments in computer algebra. They bring new algorithmic problems ranging from fundamental algebraic complexity theory to the resolution of functional equations, including algebraic and differential systems.

The challenge of conciliating reliability with efficiency and the convergence of interests for experimental mathematics and applications is an opportunity to bring together end-users of computer algebra (in mathematics, theoretical physics, mechanism design, and other applications) with computer algebra
experts. Fostering these exchanges will allow these communities to identify the different specifications, levels of certification and ways to harness computational power required by the core research of the community and in various application contexts and tackle the different complexity models.

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