Cordian Riener

Cordian Riener

Professor of Mathematics

UiT The Arctic University of Norway, Tromsø

I work in real algebraic geometry, polynomial optimization, symbolic computation, and computational complexity — and, running through all of it, on how symmetry can be turned from a feature of a problem into the very thing that makes it solvable.

Upcoming Polynomials in Tromsø (PIT 2027) — a conference on stability, hyperbolicity, and the geometry of polynomial roots, 10–14 May 2027.

Research

Positivity & sums of squares

When is a real polynomial non-negative, and when can this be certified algebraically? Cones of positive forms, Positivstellensätze, and the geometry that separates non-negativity from sums of squares.

Symmetric semi-algebraic geometry

The topology and geometry of real varieties and semi-algebraic sets invariant under reflection groups — equivariant Betti numbers, Vandermonde varieties, orbit spaces, and Specht ideals.

Polynomial & semidefinite optimization

Moment and sums-of-squares relaxations for polynomial problems, and symmetry reduction as a tool to shrink them — from crystallographic symmetries to problems in discrete geometry.

Symbolic computation & complexity

Algorithms for real solving and connectivity that exploit invariance, and the complexity-theoretic side of algebra: hardness of immanants, quadrature, and computation with symmetric input.

I am a professor of mathematics at UiT The Arctic University of Norway, where I lead the Algebra group and co-direct the Lie–Størmer Centre. I also care about the place of mathematics in culture and society, and about mathematics education at every level.

More about me →

Portrait of Cordian Riener

Selected publications

  1. The wonderful geometry of the Vandermonde map
    with J. Acevedo, G. Blekherman, and S. Debus
    Foundations of Computational Mathematics 26, 2005–2051
    doiarXivpdf
  2. Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs
    with E. Hubert, T. Metzlaff, and P. Moustrou
    Mathematical Programming 213, 517–573
    doi
  3. Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets
    with S. Basu
    Foundations of Computational Mathematics 22, 1395–1462
    doi
  4. Symmetric non-negative forms and sums of squares
    with G. Blekherman
    Discrete & Computational Geometry 65, 764–799
    doi
  5. Bounding the equivariant Betti numbers of symmetric semi-algebraic sets
    with S. Basu
    Advances in Mathematics 305, 803–855
    doi
  6. Exploiting symmetries in SDP-relaxations for polynomial optimization
    with T. Theobald, L. Jansson Andrén, and J. B. Lasserre
    Mathematics of Operations Research 38, 122–141
    doi

Full list of publications →