The seminar by Prof. Dr. Bernd Sturmfels (Max Planck Institute Leipzig) took place on 11 September 2023 in the Department of Electrical Engineering (ESAT) of the KU Leuven, Leuven, Belgium. It was the first lecture in the series Back to the Roots seminars, organized by Prof. Dr. Bart De Moor, centered around the themes of his ERC project 'Back to the roots of data-driven dynamical system identification'. The slides of the presentation can be downloaded from the seminar's webpage. Title: Gibbs Manifolds Abstract: Gibbs manifolds are images of linear 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. This is joint work with Dmitrii Pavlov and Simon Telen.