Hamiltonian Systems

Incidence theorems describe configurations of points, lines, and higher-dimensional subspaces in projective space. Two of the most fundamental examples are the theorems of Desargues and Pappus. Desargues’ theorem holds in a projective plane over any division ring, whereas Pappus’ theorem holds only in projective planes over fields. In this talk, we will uncover the topological origin of this distinction. To this end, following Richter-Gebert, Fomin, and Pylyavskyy, we will encode incidence theorems as graphs embedded in surfaces. Then, we will see that theorems associated with graphs embedded on the sphere, such as Desargues’ theorem, hold over any division ring, whereas theorems corresponding to graphs embedded on surfaces of positive genus, such as Pappus’ theorem, typically — though with some notable exceptions — hold if and only if the ground ring is a field. This talk does not assume any prior knowledge beyond undergraduate mathematics.