But you're still hungry.

My attempt at explaining what I do to a non-mathematician:

We (by which I mean everyone, not just mathematicians) are interested in symmetries of the world around us, e.g. ways to order a finite set of letters or the symmetries of shapes (equilateral triangle, cube, circle, etc.). When you divorce symmetries from the concrete object they act on and study them as abstract symmetry groups, you get abstract algebra. This is useful – for example, \(S_3\) is both ways of ordering 3 letters and the symmetries of an equilateral triangle. Representation theory is about “putting back” the concrete object and trying to collectively study “all the different ways” a group of symmetries can act on \(n\)-dimensional space (called the representations of this group), giving you a systematic way to describe any action, which is really important because the only way these abstract symmetries show up in places is when they act on things! For example, character tables in chemistry come from realizing that molecules have symmetry groups, then using representation theory to describe how a symmetry group acts on a molecule. The mathematics underlying the standard model in physics is from representation theory of what mathematicians would call \(\text{SU}(3) \times \text{SU}(2) \times \text{SU}(1)\) (don't ask me how the physics works, I don't actually know). Moreover, there is actually a formal theorem that expresses the idea that if I told you about all the representations of a group (but I never told you what the group was), you could recover the group from this description.

Now let's try something abstract: what I've described is groups acting on vector spaces (i.e. Euclidean space). What if we could replace “vector spaces” with “a family of things that behave similarly to vector spacess but aren't vector spaces” and “group” with “whatever the correct analogue of group would be in that setting”? I didn't make this up – the physicists did when they started talking about supersymmetry groups and supervector spaces! Then we can try to understand this new setting and how it is similar to and different from our more classical examples. When I say “a family of things that behaves similarly to a vector spaces but aren't vector spaces”, a more formal term for this is a (symmetric) tensor category, and one can also define what a “group” would be in such a symmetric tensor category. Why would we be interested in studying these generalizations besides finding them intrinsically interesting (though I will admit I mostly find them intrinsically interesting)?

Well, it turns out that it's representation theory all the way down (or up). Over the complex numbers, under some mild conditions (“moderate growth”), every symmetric tensor category is secretly “the family of representations of a supergroup”. (If you remove those conditions, you get very interesting new categories, including ones that are somehow morally “representations of permutations of \(\pi\) letters or \(-2\) letters or \(3.5 – i\) letters” despite not being actually able to describe what said permutations would be.) This is a crazy statement – we've defined some new class of objects which, a priori, should be this new unexplored land, but it turns out to study them, all you need to do is understand supergroups in supervector spaces. I mean, that is hard, but it's something mathematicians and physicists have been doing for a while.

Now let's try working over a “field of characteristic \(p\)” for prime \(p\) (this is a set where you can do the same operations as you can on real or complex numbers, except you specify that \(1 + 1 + \cdots + 1\), where 1 is added \(p\) times, is 0, which means the set is strange, but it's important in number theory and cryptography and stuff). Generally, representation theory is harder over said fields. We don't know how to describe symmetric tensor categories of moderate growth in this setting, but we do have a guess – they are all secretly “the family of representations of an algebraic group in the higher Verlinde category”, where the higher Verlinde category is some massively complicated, very abstract, construction that no one really understands yet. So it would sure be great if, just like how we've already developed representation for supergroups, we could do the same thing here. And that's really hard. We have to take all these notions we take for granted in the vector space case (including notions from related fields which took centuries to develop) and figure out how they work here (or if they don't), and they all end up having super abstract definitions even when they do work. We also don't know if those definitions are the “right” ones (“right” will only really be indicated by posterity, if those definitions are able to lead to our guess being proved correct and other results following). It's all very strange and cool.

So what do I do? I study a specific higher Verlinde category called \(\text{Ver}_4^+\). It's nice, in that it has a concrete definition. It's also not nice, in that several things about this category break assumptions we take for granted classically. There's two reasons we would want to care about it:

1. In this big program I described in the previous paragraph, \(\text{Ver}_4^+\) provides a prototype to conjecture what the right definitions and theorems should be in the general case. Roughly, this is because going from vector spaces (or supervector spaces) to \(\text{Ver}_4^+\) is about having things go from working to breaking (hard to fix), whereas going from \(\text{Ver}_4^+\) to other higher Verlinde categories is about things breaking even more than they already do (hopefully not as difficult to fix).
2. It's of intrinsic interest because another way to describe this category is “supervector spaces in characteristic 2”. That is, if you naively try to define supervector spaces in characteristic 2, you just get vector spaces (which isn't bad, it's just means you don't get something new to study), but you can also try to define it in such a way that \(\text{Ver}_4^+\) pops out, and then you can try to explain how representation theory in this category is like if you “replaced \(p\) with \(2\)” in the usual \(p \ge 3\) case.

What's going to come out of this? I have no idea. Maybe it'll just be a cool thing to study. But hopefully, it will allow us to broaden our perspective even more – new abstractions make it possible for us to answer concrete questions that previously seemed opaque by providing a new framework for our thinking, like how navigating a city via written directions (“turn left at X Street”) becomes much easier when you draw a map from a birds-eye view and plot your entire route on it. And after that, who knows? One day, perhaps college students will get a nicely packaged version of all this work as an undergraduate class, not as a frontier of mathematical research, and I'd be excited to see what new things people are thinking about then.