Preserving and Exploiting Symmetry in Algebraic Computation

Abstract: I wish to first present a general approach to preserving and exploiting symmetry in algebraic computations that boil down to linear algebra in finite dimensional vector spaces of polynomials. This will be illustrated on the easily understood problem of multivariate interpolation, though the underlying principle applies to global optimisation, computing cubatures, as well as computations in physics and chemistry.... The key observation is that the matrices involved become block diagonal, with repeated blocks, in a symmetry adapted bases.

As the dimension of the vector spaces of polynomials grows the computation of symmetry adapted bases becomes a bottleneck. But they are formed of fundamental equivariants. These are equipped with a finitely generated module structure over the invariant ring. This structure is to be exploited to provide a generative approach to the formation of symmetry adapted bases.

As a transition we shall observe that the symmetry preserving ideal interpolation scheme we arrive at can be directly applied to compute fundamental invariants and equivariants of a reflection group. These provide a generative path to symmetry adapted bases of higher degree. We then generalize the construction of fundamental invariants and equivariants to all representations of a finite group.

The material of the tutorial is extracted from articles co-authored by Erick Rodriguez Bazan and myself:

• Multivariate interpolation: Preserving and exploiting symmetry. Journal of Symbolic Computation 107:1-22 (2021) [doi:10.1016/j.jsc.2021.01.004]
• Symmetry in Multivariate Ideal Interpolation. Journal of Symbolic Computation 115: 174-200 (2023) [doi:10.1016/j.jsc.2022.08.014]
• Algorithms for fundamental invariants and equivariants (of finite groups). Mathematics of Computation, volume 91 number 337 pages 2459-2488 (2022) [doi:10.1090/mcom/3749]

Transcendental methods in numerical algebraic geometry

Abstract: Numerical algebraic geometry is the art of using numerical approximations to study complex algebraic varieties through their points, where algebraic methods instead consider the equations and the rings they define. Transcendental methods emerge from the integration of algebraic functions; they express in a simple way certain very fine properties of varieties at the cost of introducing transcendent functions. The objective of this tutorial is to show how numerical algebraic geometry can exploit transcendental methods through high precision computation of algebraic integrals.

Firstly, I will present the algorithms which make possible the practical use of transcendent methods. The three pillars are symbolic integration, numerical integration of linear ordinary differential equations, and integer relation algorithms. Then, I will show how to solve algebraic problems using transcendental methods based on very concrete examples: finding nontrivial endomorphisms of an elliptic curve, and finding nontrivial curves on a quartic surface using transcendental methods.

No knowledge of algebraic geometry is required.