The Mickey mouse theorem assures that a connected positive curvature graph of positive dimension is a sphere.
The parametrized poincare-hopf theorem allows to see the f-vector of a graph in terms of the f-vector s of parts of the unit spheres of the graph.
We prove that any discrete surface has an Eulerian edge refinement. For a 2-disk, an Eulerian edge refinement is possible if and only if the boundary length is divisible by 3
Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on “On Helmholtz free energy for finite abstract simplicial complexes”.) As I will have little time during the rest of the semester, it got thrown out now. It is an interesting story, relating to one of the greatest scientist, Hermann von Helmholtz (1821-1894). It is probably one of the … ….
This is a research in progress note while finding a proof of a conjecture formulated in the unimodularity theorem paper.
The proof of the unimodularity theorem is finished.
The following picture illustrates the Euler and Fredholm theme in the special case of the prime graphs introduced in the Counting and Cohomology paper. The story there only dealt with the Euler characteristic, an additive valuation (in the sense of Klain and Rota). Since then, the work on the Fredholm characteristic has made more progress and is now understood. The … ….