### Four New Talks

#### Posted by Tom Leinster

In October I did little but talk. Five talks in five locations in 23 days, with only one duplicate among them, left me heartily wishing not to hear my own voice for a while.

Having gone to the effort of making slides, I might as well share them publicly. All the talks are on topics that have come up on the Café before. Here they are:

*Unexpected connections*This was a quickie for students beginning math PhDs at Scottish universities. I was trying to inspire them with the joys of a broad education. (Beforehand, I asked for your help on what to say.)*The mathematics of biodiversity*This was a public lecture — my first ever. It was held under the banner of Mathematics of Planet Earth 2013.*The eventual image*This grew directly out of a couple of posts here at the Café. I still have the feeling that I haven’t got to the heart of the matter.*The many faces of magnitude*A subject I’ve written about extensively here, all parcelled up for a couple of colloquia.

## Re: Four New Talks

With regards to your Eventual Image talk, I’ve been trying to use categories like the $R$ given below and wonder how your analysis applies to it and whether other categories with rooted objects behave in a similar fashion (and if $R$ is in the literature).

For $p$ a pointed set and $f$ an endofuction on $p$, say that $f$ is

rootedif every element of $p$ can be reached from its point by applications of $f$. A rooted endofunction can be represented as a lasso with a leader of length $n$ and loop of size $m$ - a pair $(n, m)$The category $R$ with objects being all rooted endofunctions on all pointed sets (pairs $(p, f)$ or $(n, m)$) is a lattice where:

$\bot = (1, id) = (0, 1)$

$\top = (N, succ) = (\infty, \infty)$

$(n_{1}, m_{1}) \vee (n_{2}, m{_2}) = (max(n_{1},n_{2}), lcm(m_{1}, m_{2}))$

$(n_{1}, m_{1}) \wedge (n_{2}, m_{2}) = (min(n_{1},n_{2}), gcd(m_{1}, m_{2}))$

$R$ has atoms. $(1, 1)$ is one, as are all objects $(0, p)$ where $p$ is prime. $R$ is not atomistic (all objects cannot be given as a join of atoms) because it also has the semi-atoms:

$(n, 1)$ for $n\geq 2$

$(0, m*p)$ for prime multiples.

I guess that $R$ is semi-atomistic.