Geometry of Delta Sets

We look at a few very simple examples in this category of geometric structures. It is already now very useful as I can work with relatively small geometric representations of quite elaborate manifolds. Delta sets for example are much more natural than CW complexes which have the annoying feature that they are dynamic in that one has to store how the structure is built up. From a computer science point of view it is much simpler to just work with a set, a Dirac matrix and a dimension vector. Actually working with this is not more complicated than working with simplicial complexes. Things are also natural as any of these structures naturally comes with a finite classical topology. All this of course is completely finite. We would not touch in this context infinity. If real or complex numbers are mentioned, then we could always go with finite subsets. The real litmus test whether a structure is good is when looking at theorems. In this case, classical theorems are the same. A continuous map on a structure with trivial cohomology for example has at least one fixed point. The curvature becomes in the continuum the curvature defined using tensor calculus. There are still many battles to be faught: examples are when looking at dynamical systems on such a structure or how to define sectional curvatures in a natural way.
Pictures were taken this beautiful morning at the Charles in Cambridge. Was glad to be early as it closed up a bit later in the day.