Glass Theorem
In 1973 Leon Glass proved a discrete Poincare Hopf theorem for directed graphs embedded in a 2 dimensional manifold. Kate Perkins has related this to a discrete Poincare Hopf theorem of mine. This is a discussion of the connection.
Category: Finite topology
Spectral Monotonicity for the Hodge Laplacian
We prove that the spectrum of the Hodge Laplacian dd* +d*d depends in a monotone way on the simplicial complex.
A multi-particle energy theorem
A finite abstraact simplicial complex or a finite simple graph comes with a natural finite topological space. Some quantities like the Euler characteristic or the higher Wu characteristics are all topological invariants. One can also reformulate the Lefschetz fixed point theorem for continuous maps on finite topological spaces.