Hamiltonian Systems

In 2023, Fomin and Pylyavskyy introduced a method, which they called the master theorem, that uses quadrilateral tilings of closed oriented surfaces to find and prove incidence theorems in projective geometry. This method can be used to prove various classical results such as Desargues' theorem and Pappus' theorem. Many questions were left open by their work, however, including the question of which incidence theorems can be proved using a surface of a given genus.

For any incidence statement, we give an intrinsic characterization for the minimal genus of such a realizing surface, thus showing that they can be promoted from a feature of a proof to an invariant of the incidence statement being proved. This leads to a genus stratification of incidence theorems according to their proof topology. Using different conventions of proofs allowed, we obtain the proof-topological genus stratification at three levels.