A lot of the maths I do involves staring at pictures for a long time, this is a post about some pictures I was staring at before Christmas!
New research by Nathalie Wahl
gives a cool method that generates a so called ‘simplicial complex’ from a ‘homogenous category’ when trying to prove a result called ‘homological stability’. Forgetting what these mathsy words mean (we don’t need to know for the sake of this post) let’s look at some pictures of the ‘complexes’ generated in an example we will call the e.g. example!
The zero dimensional simplicial complex
The one dimensional simplicial complex
A representation of the two dimensional complex
The two dimensional complex can also be viewed as an octahedron (this is the same picture!)
A representation of the three dimensional simplicial complex
In each of these cases, some vertices (dots) and edges (lines) are being generated. The edges go between vertices that have different numbers, and the ‘dimension’ is one less than the maximum number we see next to a vertex.
Mathematicians have worked with simplicial complexes for many years. Therefore lots of results are known about them. Here are some pictures of normal shapes being made into simplicial complexes, which are made up of dots, lines, triangles and higher dimensional versions of triangles, such as triangular pyramids.
In our example, we can consider each picture as a collection of these triangular building blocks. When we see three vertices with different numbers, they are all joined to one another and we want to imagine that as a coloured-in triangle! You can see this a bit better in the picture where the complex is drawn as the octahedron. Similarly in the last picture we get four vertices with different numbers all joined to one another, and we want to imagine this as a solid triangular pyramid.
Simplicial complexes drawn like this (where you see dots and lines and have to imagine the higher dimensional triangles) are called flag complexes of graphs.
In my case, we want to prove a ‘high connectivity axiom’ for the simplicial complexes, and then Nathalie Wahl’s paper proves the result!
Below are some images I sketched when working things out, be warned they aren’t all correct!