## April 19, 2023

### Bargain-Basement Mathematics

#### Posted by John Baez

The fundamental theorem of Galois theory and that fundamental theorem of algebraic geometry called the Nullstellensatz are not trivial, at least not to me. But they both have cheaper versions that really are. So right now I’m pondering the difference between ‘bargain-basement mathematics’ — results that are cheap and easy — and the more glamorous, harder to understand mathematics that often gets taught in school.

Since I’m talking about bargain-basement mathematics, I’ll do it in an elementary style, at least at first — since I want beginners to follow this! I hope experts will look the other way.

## April 17, 2023

### Metric Spaces as Enriched Categories I

#### Posted by Simon Willerton

Last November I gave a talk entitled “Looking at metric spaces as enriched categories ” at the African Mathematics Seminar at the invitation of Café regular Bruce Bartlett. You may remember that John gave a seminar the month before me.

The talk was aimed at general pure mathematicians, with my main assumption being that the audience knew what a group action, a representation, a metric space and a category were.

My talk was in two halves, the first half was about enriched categories in general and how metric spaces can be viewed as enriched categories. The second half was about ‘applications’ of this that I’d been involved in, that was mainly an overview of the theory of magnitude of metric spaces and a little bit on directed tight spans because of the involvement of some African mathematicians.

I decided that I would write up the first half of the talk and that is what this post and the next post will be on. The definition of enriched category will come in the next post. Hopefully it is clear from the above that these two posts are likely to be rather basic as far as the regulars at the Café are concerned!

## April 12, 2023

### Eulerian Magnitude Homology

#### Posted by Tom Leinster

*Guest post by Giuliamaria Menara*

Magnitude homology has been discussed extensively on this blog and definitely needs no introduction.

A lot of questions about magnitude homology have been answered and a number of possible application have been explored up to this point, but magnitude homology was never exploited for the structure analysis of a graph.

Being able to use magnitude homology to look for graph substructures seems a reasonable consequence of the definition of boundary map
$\partial_{k,\ell}$. Indeed, a tuple $(x_0,\dots,x_k) \in MC_{k,\ell}$ is
such that $\partial_{k,\ell}(x_0,\dots,x_k)=0$ if for every vertex $x_i \in
\{x_1,\dots,x_{k-1} \}$ it holds that $len(x_{i-1},\hat{x_i},x_{i+1}) \lt len
(x_{i-1},x_i,x_{i+1})$. In other words, if every vertex of the tuple is
contained in a *small enough substructure*, which suggests the
presence of a meaningful relationship between the rank of magnitude
homology groups of a graph and the subgraph counting problem.