Symmetry, in its most basic mathematical sense, is invariance under the action of a group. The premise of SymRAG was that this simple idea has real consequences for polynomial systems: the symmetries of a problem can be used to describe the geometry and topology of the sets it defines, to design faster algorithms, and to make optimization problems tractable – and, in some cases, they are precisely what makes a problem hard. The main lines of work are summarized below; full references are on the Publications page.

Symmetry in algebraic structures

Much of the project concerned ideals and varieties invariant under the symmetric group S_n or the hyperoctahedral group. A central object were the Specht ideals – ideals generated by Specht polynomials – whose structure was described in detail and used to solve symmetric systems of polynomial equations (Moustrou–Riener–Verdure; Debus–Moustrou–Riener–Verdure). A related line of work gave explicit descriptions of real orbit spaces of finite groups by few inequalities (Moustrou–Riener–Schabert) and of orbit spaces of Weyl groups acting on compact tori (Hubert–Metzlaff–Riener).

Symmetry and topology

In a series of papers with Saugata Basu, symmetry was brought to bear on the homology of symmetric semi-algebraic sets: Vandermonde varieties and mirrored spaces give a structural description of their cohomology, and the multiplicities of the irreducible S_n-representations that occur are bounded polynomially in the number of variables. The later work with Acevedo, Blekherman and Debus on the geometry of the Vandermonde map and on symmetric nonnegative functions continues this theme.

Infinitely many variables

Symmetric problems have a natural limit as the number of variables grows. With Mario Kummer, the project developed an equivariant algebraic and semi-algebraic geometry of infinite affine space, extending the classical picture to this infinite-dimensional setting and opening a number of new questions.

Symmetry as an algorithmic tool

A second strand treated symmetry as something an algorithm can exploit. The project produced new algorithms for real root finding and for computing critical points of invariant algebraic systems (Riener–Safey El Din; Faugère–Labahn–Safey El Din–Schost–Vu; Labahn–Riener–Safey El Din–Schost–Vu), and an algorithm for deciding connectivity in symmetric semi-algebraic sets (Riener–Schabert–Vu), which earned Robin Schabert the best PhD paper award at ISSAC 2024.

… and as a source of hardness

Symmetry does not always help. The work with István Miklós on immanants – the family of matrix functions interpolating between determinant and permanent – shows that the symmetries built into these functions are partly responsible for their #P-hardness, even on restricted classes of matrices.

Symmetry in optimization

The most applied part of the project used symmetry reduction in polynomial optimization: for trigonometric polynomials with crystallographic symmetry, with applications to spectral bounds for set-avoiding graphs (Hubert–Metzlaff–Moustrou–Riener), for AM/GM-based (SAGE and SONC) certificates (Moustrou–Naumann–Riener–Theobald–Verdure), and for reflection groups and cones of sums of squares (Debus–Riener). A survey of the field, Symmetries in polynomial optimization, appeared in the volume Polynomial Optimisation, Moments, and Applications.

Beyond symmetry

The project also worked on quadrature rules with few nodes (Riener–Schweighofer; Blekherman–Kummer–Riener–Schweighofer–Vinzant), on the real evolute of plane algebraic curves (Piene–Riener–Shapiro), and – through the LAUMA initiative – on mathematics education, with a study of attitudes towards mathematics among STEM and non-STEM students in Norway (Óturai–Riener–Martiny).