  Gibbs manifolds are images of affine spaces of symmetric matrices under the
exponential map. They arise in applications such as optimization, statistics
and quantum~physics, where they extend the ubiquitous role of toric geometry.
The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs
manifold. We compute these polynomials and show that the Gibbs variety is
low-dimensional. Our theory is applied to a wide range of scenarios, including
matrix pencils and quantum optimal transport.
