Profound but not popular mathematical topics and notions

Question

The algebraic Theory of Invariants used to be a hot topic until David Hilbert proved his two theorems about invariants. Then for tens of years, the popularity of the topic went down a long time before it picked up again.

Question What are today's mathematical known topics and/or notions that are profound but not popular? Together with an example, could you add an explanation of such a situation?

Example from Computer Science -- geometric SIMD (fine grain parallel processing) was a popular and hot topic till the middle of 1980'. Then you hardly hear about it while the idea is as fundamental as always.

The explanation is two-folded but very simple. On the one hand, there is some learning and new understanding involved in geometric SIMD processing; one needs to acquire new habits, new reflexes. On the other hand, the technology progress was such that people were satisfied with the results obtained without bothering with the SIMD ways. (Underneath, the new traditional computer architecture is not that traditional -- these days, it incorporates quite a bit of parallelity). We see that the geometric SIMD is not popular for the wrong reasons, and a lot of potentials is wasted.

Answer 1

This hasn't yet been revitalized, but I think John von Neumann's work on Continuous Geometry is rather deep, but there really doesn't seem to be much major work on this topic besides what you see in the references in the link above.

Oddly, even though von Neumann was explicitly aiming to de-emphasize the notion of point in geometry via this work, and the axioms for a continuous geometry are quite similar to those of the notion of frame in the theory of locales, when I have read about locales I have never seen von Neumann's work referred to as a precursor to the theory. (Of course frames only require finite meets.) I'm surprised about this since undoubtedly Marshall Stone was involved with the prehistory of locale theory as is reflected in Elements of the History of Locale Theory by Peter Johnstone

I've seen it mentioned on MO that during the East Coast Operator Algebra Symposium a while back concentrated on the $\mathbb{F}_{1}$ approach to RH, Alain Connes outlined how von Neumann's continuous geometry may have something to say about this approach. In the subsequent years, of course, Connes and Consani have found the Arithmetic Topos...

This might be opinionated, as feared, but I'd be interested in knowing what happened to this idea of von Neumann over the years, and how one can trace it through the literature (I'd like a lead...beyond von Neumann's text...)

Answer 2

Category theory - though I should add, I'm no expert as experts go.

It's often thought, according to what I've read, as some fearsome and formidable machine - and I'd agree with that seeing some of the texts I've seen. But almost everything is when built over a long period, with care, and dedication.

We teach arithmetic to school-children but not higher number theory even though they rely on the same framework: the integers.

Likewise, if I had to teach category theory to kids or lay adults; I'd simply say they are 'curved' vectors (expecting them to know what vectors are). Most likely, they would think it simple to be profound ... but life and thought is often like that.

Another one might be graph theory. It's readily understandable what a graph is, easily drawable; though all the books, I've seen eschew the visual and give a really dry and axiomatic description. This really annoys me. There is nothing wrong with explaining concepts with drawings.

Finally another, for me, would be differential geometry. On the usual exegesis it's very difficult to understand. But it's easily understandable by drawing diagrams and this links into the first profound idea - that is category theory - since it's easy to motivate the tangent functor this way (if not the cotangent bundle, but none cannot have everything).

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@Wojuwu: Well, for one thing, there is no such tag on MathOverflow. Does that answer your question? For another, the question asks for concepts which are profound but not popular. Teaching ideas to school-kids, seems to exemplify this. We teach kids, big ideas in an easy way ...