This presentation is getting into the idea of "magnitude" the invariant from category theory and how it can be used in mathematical ecology, especially when we're trying to max out diversity. It's put forward as a possible answer to a head-scratcher in mathematical ecology: How do we find the maximum diversity if we're given a list of n species and a similarity matrix?

The presentation gives us a whole range of views on biodiversity, making it clear that we need to think about both common and rare species when we're figuring out diversity. It hints that this magnitude idea might give us a more detailed picture of diversity by considering how important different species are relative to each other. Then the presentation gets into the weeds of category theory, chatting about enriched categories and size-like invariants. It talks about stuff like monoidal categories, V-enriched categories, and linear categories. These ideas are brought up as tools to help us understand and calculate magnitude.

The presentation also explores how these category theory ideas relate to metric spaces. It suggests that getting a handle on this relationship could give us even more insight into how to max out diversity. Lastly, the presentation brings up the Euler characteristic, which is a concept from algebraic topology. It hints that the Euler characteristic and magnitude are pretty tight, and understanding this connection could give us even more insight into how to max out diversity.

So, to wrap it up, the presentation is a thorough look at how category theory ideas, especially magnitude, can be used to tackle problems in mathematical ecology. It suggests that these ideas can give us a more detailed understanding of diversity and might even help us figure out how to get the most diversity.

References:

https://arxiv.org/abs/2012.02113

https://arxiv.org/abs/1606.00095