This might be useful in relatively young fields (e.g. information theory) , or ones where the topic itself thwarts any serious tower-building (e.g. archaeology), but in general I think this is likely to be misleading. Important problems often require substantial background to recognize as important, or even as problems.
A list of applications of sophisticated math to physics which is aimed at laymen is going to end up looking a lot like a list of applications of sophisticated math to astrophysics, even though condensed matter is an order of magnitude larger, and larger for sensible reasons. Understanding topological insulators (which could plausibly lead to, among other things, practical quantum computers) is more important than understanding fast radio bursts (which are almost certainly not alien transmissions). But a “fast radio burst” is literally just a burst of radio waves that appeared and disappeared really fast. A neutron star is a star made of neutrons. An exoplanet is a planet that’s exo. A topological insulator is … an insulator that’s topological? Well, A: no, not really, and B: now explain what topological means.
Once people can ask well-posed questions, providing further motivation is easy. It’s getting them to that point that’s hard.
Here’s another approach, using tools that might be more familiar to the modal reader here:
This sort of “forwards on space, backwards on stuff” structure shows up pretty much anywhere you have bundles, but I often find it helpful to focus on the case of polynomial functors—otherwise known as rooted trees, or algebraic data types. These are “polynomials” because they’re generated from “constants” and “monomials” by “addition” and “multiplication”.
Viewed as trees:
The constant
A * x^0
is an A-labelled tree (all our trees have labelled nodes and are furthermore possibly labelled as trees, and the same is true of all their subtrees) with no nodes.The monomial
A * x
is an A-labelled tree with one x-labelled node.(F + G) x
joinsF x
andG x
by adding them as children of a new root node(F * G) x
joinsF x
andG x
by merging their rootsViewed as data types:
The constant
A * x^0
isA
The monomial
A * x
is(A, x)
(F + G) x
is the disjoint union ofF x
andG x
(F * G) x
is(F x, G x)
The tree view gets messy quick once you start thinking about maps, so we’ll just think about these as types from now on. In the type view, a map between two polynomial functors
f
andg
is a function∀ x . f x -> g x
. Well, almost. There are some subtleties here:That’s not quite the usual set-theoretic “for all”: that would be a recipe that says how to produce a function
f x -> g x
given any choice ofx
, but this is one function that works for every possiblex
. This is usually referred to as “parametric polymorphism”.It’s actually not quite that either: a parametrically polymorphic function isn’t allowed to use any information about
x
at all, but what we really want is a function that just doesn’t leak information aboutx
.But we’re just building intuition, so parametric polymorphism will do. So how do you map one tree to another without knowing anything about the structure of the node labels? In the case where there aren’t any labels (equivalently, where the nodes are labelled with values of
()
, the type with one value), you just have an ordinary functionspace: f () -> g ()
. And since this determines the shape of the tree (if you’re hesitating here, working out the induction is a good exercise), all that’s left is figuring out how to fill in the node labels.How do you fill a
g
shaped container with arbitraryx
values, given only anf
container filled withx
values? Well, we don’t know how to producex
values, we don’t know how to combinex
values, and we don’t know to modifyx
values, so the only thing we can do is take the ones we already have and put them in new slots. So for every slot in our outputg ()
that we need to fill with anx
, we pick a slot in each input tospace
that can produce it, and commit to using thex
it contains. This is the backwards part of the map.There is a deep connection between Chu spaces and polynomial functors by way of lenses, but I don’t fully understand it and would rather not risk saying something misleading. There is also a superficial connection by way of the fact that Chu spaces and polynomial functors are both basically collections of labelled points put together in a coherent way, which I hope is enough to transfer this intuition between the two.