Martin Hünniger:Axiomatic discrete geometry and the Jordan-Brouwer theorem

Martin Hünniger:Axiomatic           discrete geometry and the Jordan-Brouwer           theorem
10/01

2009. október 01.

Déli tömb 2.211

10/01

2009. október 01. -

Déli tömb 2.211


Abstract The geometries used in digital geometry differ greatly from the euclidean geometries. For instance on a line represented by a computer there are only finitly many points between two given points. It is possible to give an axiomatic characterisation of these geometries for arbitrary dimensions. It turns out, that the standard model for this axiomatic system is the n-dimensional lattice Zn. By further investigation one can see that not every Jordan-curve in Z2 fulfills Jordans theorem. Similiar issues arise in higher dimensions. Even worse, the definition of a higher dimensional discrete manifold is complicated. To establish these notions in discrete geometry, methods of combinatorial topology are applied. In this talk I will give an overview on the results of my PhD Thesis. These results include the axiomatisation of the arbitrary dimensional discrete geometry, the definition of the discrete manifold and the transformation of the Jordan-Brouwer theorem in the discrete setting.