The Euler characteristic counts alternating dimensions of a topological space: vertices minus edges plus faces minus higher-dimensional analogs. It is a topological invariant — preserved under continuous deformation, indifferent to coordinates.
The maximum likelihood degree counts something entirely different: the number of critical points of a likelihood function on a statistical model. It measures the algebraic complexity of maximum likelihood estimation — how many candidate solutions exist before you pick the best one.
Hosten et al. prove that for hypersurfaces in products of projective spaces corresponding to three-way independence models, the signed Euler characteristic equals the ML degree. A topological invariant — a number that knows about shape — is exactly equal to a statistical invariant — a number that knows about estimation complexity.
The equivalence means that counting holes and counting likelihood critical points are the same operation in disguise. The topology of the parameter space determines the algebraic difficulty of fitting the model. A topologist measuring the shape of the model space and a statistician counting the solutions to the score equations arrive at the same number.
This is not an analogy. It is an identity. The geometry of the space dictates the complexity of inference in it.