The goal of this talk is to explore what symmetries can say about an object. We will then focus on the case of algebraic varieties, where the symmetries are birational self-maps. This is joint work with L. Esser, A. Regeta, C. Urech, and I. van Santen.

A key problem in computer proofs based on solutions from copositive optimization, is checking whether or not a given quadratic form is completely positive or not. In this talk we describe the first known algorithm for arbitrary rational input. It is based on a suitable adaption of Voronoi's Algorithm and the underlying theory from positive definite to copositive quadratic forms. We observe several similarities with the classical theory, but also some differences, in particular for three and more variables. A key element and currently the main bottleneck in our algorithm is an adapted shortest vector computation, asking for all nonnegative integer vectors attaining the copositive minimum of a given copositive quadratic form. (This is based on joint work with Valentin Dannenberg, Alexander Oertel, Mathieu Dutour Sikiric and Frank Vallentin).

The rank $n$ superspace $\Omega_n$ is the algebra of polynomial-valued differential forms on affine $n$-space. This carries an $G$-action for any pseudo-reflection group $G$ -- two important examples being the symmetric group $\mathfrak S_n$ and the hyperoctahedral group $\mathfrak B_n$. The superspace coinvariant ring for $G$, defined as the quotient of $\Omega_n$ cut out by $G$-invariants of $\Omega_n$ with vanishing constant term, has received increased attention in recent years. In this talk, we explore some recent results on the superspace coinvariant rings for $\mathfrak S_n$ and $\mathfrak B_n$, including their Hilbert series, explicit monomial bases, and their representation-theoretic structures.