Toric varieties and Gibbs Manifolds in Convex Optimization

Entropic regularization for linear programming leads to intersecting a toric variety with the feasible polytope. In semidefinite programming, the toric variety is replaced by a new geometric object, called Gibbs manifold, and the feasible polytope becomes a spectrahedron. I will explain these concepts and present the example of (quantum) optimal transport. This is based on joint work with Dmitrii Pavlov, Bernd Sturmfels, François-Xavier Vialard and Max von Renesse.