Binomial ideals are important objects in algebraic statistics. A frequent question is whether a given family of binomial ideals stabilizes up to symmetry when some of the parameters grow unboundedly. If this is the case, then the complexity of these models is reduced, therefore computations with them can be carried out more efficiently.

In this project we study stabilization up to symmetry. We aim to find a detailed way to work with families of toric varieties with up to symmetry finite Markov bases and to extend the understanding of the polyhedral cones and lattice point configurations modulo symmetries. We envision efficient theoretical and algorithmic methods resulting from a better understanding of how to deal with polyhedral objects modulo symmetry.