Are there any mathematics that could only be learned by very few people?

I'm worldbuilding a situation where new drug increases IQ when given to preschool children. Unfortunately only 1% of the recipients get the benefits, the rest risk their development being stunted. As a result, the drug is illegal. This creates a moral dilemma for the parents: try your luck or play it safe.

I need some kind of mathematical discipline that could be understood only by very few gifted people, and preferably studied at postgraduate levels . This discipline serves as a device to show how much drug improves the mind of the children, and that normals can't hope to compete.

Is there anything like that ?

I'm looking for a discipline that exists. My plan is to watch MOOC and learn enough to be dangerous, then have real mathematician to review my ideas.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monty Wild Oct 28 '19 at 5:50
• The difficulties often come from that fact there is no appropriate learning materials for not-math-talented people about anvanced/frontier math, which is mostly based on papers and books written by math professionals for professionals which focus more on the process of definitions and proofs. Not being a math-talent, you get lost easily based on thess\e materials. With a book that focus on concept/motivation/the intuitive relations, they will be more accessible for more not math-talented people. – jw_ Oct 28 '19 at 6:57
• en.wikipedia.org/wiki/Savant_syndrome I think you're knocking on the wrong door. Oddly enough, an autism-inducing drug seems to fit your bill (mostly). – Mephistopheles Oct 28 '19 at 15:08
• If Facebook is anything to go by, I'd say order of operations... :) – Frauke Oct 28 '19 at 15:49
• For me %1 percent is a very small probability, to make my probably normal child a little bit smart or make them quite stupid. – atakanyenel Oct 28 '19 at 16:48

I have a PhD in Mathematics and came across this question. To be honest, I dislike almost every single answer, except maybe L.Dutch's answer concerning Wile's proof of Fermat's last theorem. However, I do think there is a much, much better candidate, and one that would make every mathematician reading your story quite delighted:

https://en.wikipedia.org/wiki/Langlands_program

Quoting (and I agree):

Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as “a kind of grand unified theory of mathematics.”

The good thing about the Langlands program is that it is not (yet) a worked-out theory and a topic of active research, with only very few, extremely talented mathematicians being able to make contributions, like Peter Scholze. It seems to fit your story like a glove; people who received the drug's benefits would be among the few who could advance the program.

I encourage every reader to also look at the comments that were moved to the chat, it adds a lot of value. Among other things, a very important point is brought up about how my phrasing is arguably hyperbolic: There are certainly more than just a handful of people who have contributed to the program. However, I still think it is very few people, when compared to all of mankind and even to all of mathematics.

I would also like to address a question from the comments, namely whether I can give a brief summary of the Langlands program. To be completely honest, the answer is "no". I have studied mathematics for 10 years and some of it was spent in related fields (algebraic geometry and quite some representation theory as well, especially algebraic groups) - but I still do not feel qualified to give a reasonable summary of the Langlands program, let alone one that would be comprehensible to a layman. I do have an idea of what it is about, but I struggle to put that into words that do not demand quite advanced material. Have a look at the wikipedia entry, my honest summary would probably be quite similar to it. I do not understand it well enough to also explain it well. But this is the point - I don't think many mathematicians do.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monty Wild Oct 28 '19 at 5:53
• Setting aside the issues discussed earlier, one of the problems of putting Langlands into a story is that, well, very few people can competently describe what it is. We see here that even trained mathematicians outside the subject struggle to summarize it. And if you (assumed to be a non-mathematical author) are trying to communicate it to a non-mathematical audience, the possibility of getting anything across drops even lower. It would end up coming across as little more than a buzzword like "chaos mathematics". – Priska Oct 28 '19 at 13:20
• @Priska the problem is, everything you say will be true of any good answer to this question. – Nathaniel Oct 28 '19 at 17:01
• I likes what Eduard Frenkel said in Love and Math about it being a Rosetta Stone between the manifold defined by and equation, the elliptic curve, and the set of solutions over some prime field. – Daron Oct 29 '19 at 0:14

No such thing exists.

All mathematics is a type of language. Like language, it looks mysterious to people who don't speak it. But if you study it enough, you will understand it. There are no exceptions. (*)

Calculus was once an arcane branch of knowledge known only to Newton, Leibniz, and their handful of peers. It made them gods in terms of their ability to solve problems no one else could reach. It was the scientific nuclear weapon of its time. The closest thing the real world has to magic.

And now... we have hundreds of millions of children across the world learning calculus in school. Bookstores hawk endless numbers of texts on acing your calculus exam. This once-awesome, mysterious branch of math is now just another piece of everyday mental furniture.

The same goes with algebra, and even algebraic notation. Someday, the same will happen with all of the math known today.

(*) This means that there are no exceptional types of math where this is not true. I am not saying there are no exceptional people who might fail to understand math (brain-damaged, comatose, etc.). But the vast majority of people will understand any topic of math if exposed properly and given the right prerequisite knowledge.

• Comments are not for extended discussion; this conversation has been moved to chat. – HDE 226868 Oct 27 '19 at 3:46
• @Nosajimiki This issue has been discussed quite extensively in previous comments (now moved to chat) by myself and others. You may find the discussion there helpful. Yours being the latest of numerous comments, I will take a break from replying for now. Perhaps others will have something insightful to say. – Priska Oct 28 '19 at 17:43
• This is an empirical claim, with no actual research to back it up. But I can give personal testimony to the fact that, on many occasions, I have been stumped by a mathematical concept, purely due to the complexity of the concept and not because I wasn't familiar with any terminology or syntax. – Bridgeburners Oct 28 '19 at 20:07
• @Bridgeburners This is not even an empirical claim, it is wishful thinking by someone who has never really got close to modern mathematics or to teaching. Yet a number of people would love for this to be actually true, as you can tell by the positive balance of the votes on this answer. – Arnaud Mortier Oct 28 '19 at 20:17

Instead of looking for new math that the children can do, show them learning math faster.

All new math is built on the old. To do some incredibly complex proof, you'll typically need algebra, equations, maybe calculus or group theory or probability or what-have-you. The point is, it will be clear that these children are exceptional WELL before they invent new math, since they'll be solving systems of equations in kindergarten and integrals in first grade or whatever.

So I challenge the notion that the way to show how much the drug improves children's intelligence is to show them doing math adults can't do. It will be clear from their ability to master existing math at such young ages.

Also, if what you're going for is a drug that improves general reasoning ability, then I would find it very strange if all of the children become masters of one specific sub-field of math, like, say, chaos theory. Why should that be the case? The frontiers of modern math are in chaos theory, yes, but also number theory and complex analysis and so on. Why would all of the children become experts in one particular subject, to the extent that it becomes the de-facto test of the drug's effects?

You can represent both intelligence and mathematics more faithfully just by showing that they can do math that undergrads or grad students are doing.

• Honestly, this is the first thing I thought. Creating tests to demonstrate how quickly people can learn in a variety of subject areas would be a much better measure of intelligence. Otherwise you could have accidentally dismissed the next Beethoven or Shakespeare because they didn't solve a differential equation fast enough. Even geniuses usually aren't geniuses at everything. – DoctorPenguin Oct 28 '19 at 10:03
• This. See Star Trek:TNG, where ~10-year-old children learn calculus. youtube.com/watch?v=ETt8GJRbqLc – dissemin8or Oct 28 '19 at 15:43

Take the mathematics needed to understand Wiles's demonstration of last Fermat's theorem.

no three positive integers a, b, and c satisfy the equation $$a^n + b^n = c^n$$ for any integer value of n greater than 2.

Without a master in mathematics you cannot even think of starting to learn the basis for it.

The demonstration above is based on linking modular forms

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.

and elliptic curves.

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form $$y^2 = x^3 + ax + b$$ which is non-singular; that is, the curve has no cusps or self-intersections.

Inter-universal Teichmüller theory is a real-life example of mathematics understood only by a handful of people, nearly all of whom are students of the guy who created it. There is a claimed proof of the abc conjecture which has so far been neither verified nor definitively disproven because the material is so impenetrable.

https://en.wikipedia.org/wiki/Inter-universal_Teichmüller_theory

• This isn't a great example, A major part of the issues with IUTT is that it has been really poorly explained. Mochizuki was infamous for being not great at writing or explaining things even before his claimed proof of the ABC conjecture. Heck, I had a professor in grad school who used Mochizuki as an example of someone whose results were good enough that he was able to get away with poor writing, and then told me that I wasn't in that class. – JoshuaZ Oct 27 '19 at 15:08
• kurims.kyoto-u.ac.jp/~motizuki/SS2018-08.pdf – Priska Oct 27 '19 at 20:46
• @JoshuaZ I see, I didn't know that. I always thought it was just incredibly tough stuff. – Jesko Hüttenhain Oct 28 '19 at 7:21
• @JoshuaZ granted, but I still think it’s a fine example for the purpose of the question - if it were an easier topic to understand, perhaps it wouldn’t need great explanation, and additionally it is self-evidently a source of genuine drama! – jl6 Oct 28 '19 at 21:55

N-dimensional geometry, where n > 4. It’s very difficult for our regular human brains to cope with it, but may well have all kinds of useful implications for physics.

• countless mathematicians and physicists work everyday with spaces with dimension higher than $4$, or indeed with just general dimension $n$, if the space they're working with even has a well defined concept of dimension. It's not by any means an arcane or not well studied field. It's just plain old regular math for anybody above BSc level, because the point is that for a mathematician there's nothing special about 3 or 4 dimensions. – user2723984 Oct 26 '19 at 14:03
• But it's very difficult to have a good intuition for, and it shows a superhuman level of spatial reasoning. If you ask them, "how many square faces are in a 5-hypercube," and they answer "80" as easily as if you'd asked them how many sides are on a square, that's something nearly all mathematicians would struggle to visualize. And this is a much more suitable test for a young child, simply because it doesn't involve any mathematical formalism beyond counting and shapes. – Gilad M Oct 26 '19 at 14:13
• Why n>4? 4-manifolds are known for being ridiculously difficult to study. For example, the generalized Poincare conjecture is only unsolved in dimension 4. In dimension 4, Euclidean space has a continuum of exotic smooth structures, and it is the only dimension in which Euclidean space has any exotic smooth structure. In dimensions greater than 4, classification of manifolds becomes much easier. – Eben Cowley Oct 26 '19 at 15:23
• Also I'd like to point out that high-dimensional geometry has applications in any field involving multivariable problems, not just physics. – Eben Cowley Oct 26 '19 at 15:25
• Just because it's difficult for you to understand n-dimensional geometry does not mean that very few people can actually learn it, or even that it's particularly difficult. You just have to learn and use, you know, math, not visual intuition. – forest Oct 27 '19 at 1:16

There are a few different ways I would answer this question depending on how you actually plan to write this story. I will interpret it in a few different ways and give answers below.

Is there a field of math learn-able by only a few individuals?

No.

Another answer already pointed this out, but the vast majority of human knowledge can be understood by the vast majority of humans (if they put in the effort). The psychological literature is becoming saturated with evidence that learning and performance are more inhibited by self-consciousness than by innate intelligence (which has been shown to be flexible). Here is some research on the topic:

Trait beliefs that make women vulnerable to math disengagement

Conceptions of ability: Nature and impact across content areas

Mind-Sets Matter: A Meta-Analytic Review of Implicit Theories and Self-Regulation

Is there a field of math which very few individuals have taken the time to learn?

Yes, more than I could possibly list. People already named several examples of this in other answers, and if you'd like more examples you could go to the faculty directory of any univeristy math department, and look at what the different mathematicians are interested in.

In your question you specifically mention that you'd like a topic that is introduced in graduate courses in mathematics, not undergraduate courses. Note that topics which are introduced in undergraduate courses are still actively researched; for example, people still do research on things like integration methods.

However, I will try to address that part of your question. In the US, most undergraduate curriculums for math majors require an understanding of basic analysis and algebra, but not so much geometry or topology. In their early graduate years, students will usually be introduced to things like algebraic topology and differential geometry.

If you plan on having a character attend a class on algebraic topology or differential topology, please bear in mind that they must first understand abstract algebra and calculus, respectively. This is important regardless of the topic you choose, as you risk breaking realism for people who know how learning math works.

What is an impressive mathematical feat that would demonstrate how intelligent this drug makes kids?

This interpretation might be a bit off from your original question, but I think it might do you good to consider it. Instead of saying "and then the child could do advanced fractal chaos math" what if instead you specifically named an unsolved problem that the child solved?

You can find a long list of unsolved problems here.

I think you should really consider this approach because it is potentially more engaging for your readers.

On the one hand, you could pick a field of math that has an esoteric-sounding name and then pick an open problem in that field which seems interesting. There isn't anything necessarily wrong with this approach, the only drawback is that readers may run into a brick wall of prerequisites if they try to understand the problem that you select.

Another approach might be to select a problem that anyone can understand: Goldbach's conjecture, the twin prime conjecture, and the Collatz conjecture are all examples of famous open problems that are extremely simple to state. This way, the reader might learn something that they actually have the prerequisite knowledge to understand. This could potentially improve the reader's engagement, but the choice is ultimately up to you.

I am an assistant professor in the University of Caen-Normandy.

Some very good answers have been given already, but there is one actual branch of mathematics that has been left aside and which is one of the most arcane from my point of view (and also seems to fit your purposes quite well), that is modern algebraic geometry.

Traditional algebraic geometry is, very roughly, the study of curves or higher dimensional objects where one or more polynomials vanish (e.g. the most well-known parabola is the set of points where $$y=x^2$$, in other words $$y-x^2=0$$).

In the second half of the $$20$$th century, a man called Alexander Grothendieck had an idea of how to take this theory to a level of abstractness that would eventually make it so powerful as to radiate to neighbouring areas of mathematics (incl. most branches of topology and geometry) and generalize them as well.

The problem is, there is really no easy way to describe even the most elementary kind of object that algebraic geometry deals with, even though the people who work in that branch today will tell you that what they have in mind is nothing but "geometry". To get a grasp of it, you would have to know some abstract algebra already, and then although you might get used to the definitions and properties of these objects, chances are that you will never really get the "geometric" feeling about it that is really necessary to do anything useful in this theory.

I'm a mathematics researcher at the University of St Andrews.

Intead of focusing on a specific discipline within mathematics, I would focus on the children being able to resolve long-standing open problems. There are many problems in advanced mathematics that have been open for quite some time and are considered quite important for the development of the subject. The most prominent examples are the six remaining Millennium Prize Problems, including:

Other long-standing open problems include:

Lots of very smart mathematicans can spend decades working on one of these problems without making much significant progress. If children who were given this drug could reliably solve one of these problems in a couple of years, starting with essentially no knowledge of advanced mathematics, it would certainly demonstrate that these children were operating at a superhuman level of intelligence.

Even the idea of a child being able to make any significant progress on one of these problems would be aboslutely extraordinary.

Very large numbers.

I don't just mean numbers that are too big for a human to really understand; a billion fits that category. I don't even mean numbers too big for our usual naming convention; that caps out at 1063 (one vigintillion).

I mean numbers that make how we normally talk about numbers meaningless. Numbers you can't even write down in a way that non-mathematicians would understand (after all, any number you can fit on a whiteboard basically rounds down to zero). Graham's number is a famous example. Using our existing number system, it would have more digits than there are particles in the universe (which is about 1080)-- in fact, if you tried to count how many digits it would have, that number would have more digits than there are particles in the universe, and that number's number of digits would still have more digits than there are particles in the universe, and that pattern continues more times than there are particles in the universe. It's such an staggeringly large number that expressing it in writing requires a whole different numbering system.

This branch of mathematics has the added bonus of being very hard for computers to handle-- computations would take too long.

• As someone who’s done work adjacent to this sort of thing, it would be flattering to think that it can only be done my extremely gifted people, but I think that’s pretty implausible. Certainly the topic is well within reach of any undergraduate majoring in math with a little effort. – Henry Towsner Oct 26 '19 at 14:36
• Also, any number that can be computed precisely by a human (however bright) or computer (however powerful) will be dwarfed to insignificance by easily-defined incomputable numbers, making those computable numbers less impressive. – Thom Smith Oct 26 '19 at 16:47
• You should look up the Bekenstein bound, which is conclusive that no machine, much less human mind, could work with such large numbers. – R.. GitHub STOP HELPING ICE Oct 27 '19 at 1:01
• This answer is a bit silly, to be honest. Large numbers are actually so easy to understand that they have become a bit of a niche in popular mathematics, and there are countless Numberphile videos on them explaining them in very simple terms. – forest Oct 27 '19 at 1:24
• Even exceedingly large notations aren't that difficult to wrap your mind around. See this comment and this one where I was able to arrive at a fairly intuitive understanding of the ↑↑ operator (x↑↑y is "x raised to the power of itself" in a tower y times" just as "x↑y (or x^y) is x times itself y times") and was then able to rapidly convert between x↑↑y and 10^z. Took an hour or two. – Draco18s no longer trusts SE Oct 27 '19 at 4:23

Financial securities accounting. Please watch Limitless to see how this plays out. He can understand many different concepts very clearly and even piece them together, making him very wealthy in predicting stocks. But the drug he uses has negative consequences.

• Also in that film she becomes so smart that she picks up a child and swings the child at the baddie, so the child's ice-skated slice through the baddie's neck and he dies. – Daron Oct 29 '19 at 0:17

Category Theory

Both Wikipedia and Quanta go into some detail.

The basic idea of category theory is pretty accessible to any professional mathematician, but the details of it are not well-studied (especially "infinity categories"). However, it has applications in both foundations of mathematics as well as computer science, and is a little bit trendy in some circles.

The Quanta article explains how major results from category theory are simply quoted at an almost mystical level by mathematicians without them bothering to learn the details. It's not so much that the math is beyond the skill of all but a few mathematicians. Rather, it is too much work and not everyone is interested in learning it, even though it may be relevant to their own work. But since it affects the fundamentals of math, it may be that in the future, it is considered required foundational knowledge. That's why some folks are working to make it more accessible to mathematicians across the spectrum.

So, I would lean toward just looking at the research topics of any well known applied math organization and seeing what cool names you find. SIAM is pretty well known.

https://www.siam.org/

Why applied math? I would think of math as part of the toolkit that scientists & engineers have. We all work in the frameworks we understand. If we want to over-simplify the process, the mathematicians are already way out in front cutting a path (with a framework made for path cutting) while the rest of us are using bits and pieces cobbled together by all their old frameworks. Stuff gets simplified as we pull it out of the mathematicians toolbox so that us normies can understand it.

Ability to understand complexity seems to be as good a standin for intelligence as any. The translators from math to engineering/science need to understand the complexity on both sides to come up with a simplified bridge for the rest of us. In real life this is a group project, but maybe your super intelligent badasses can go it alone.

IMO Control Theory is a good starting point.

Also, not on the SIAM site, but Information Theory is at the intersection of a bunch of fields.

https://en.wikipedia.org/wiki/Information_theory

And if you have to, I think this might be a slightly tropey trick, but you can always combine two fields if you are just name dropping. Applied geometric uncertainty quantification. Information theoretic imaging science. Numerical life science (although I'm sure that's a thing already).

• I know enough statistics, differential geometry and information theory to be dangerous. Though they all require some affinity & work ethic none of them is hard enough to be reserved for cognitive elite. At high level the situation is probably different. – Epipihany Oct 25 '19 at 19:57
• I don't think fields that can only be worked in by the cognitive elite exist. They're sprinkled about randomly. IMO progress is the process of taking tasks that required genius and making them things that we can all do. Affinity & hard work put you in the elite. Lucky timing puts you into the history books. – Zwuwdz Oct 25 '19 at 20:20

The mark of a good mathematician is less what she can understand, more what she can explain. For example, Einstein was able to simplify the understanding of the entirety of energy and matter down to 5 symbols. Leibniz et al. simplified a huge amount of engineering and mathematics down to a single number.

One of the most fundamental problems we have in AI is understanding that mathematical ability or IQ are not the same as intelligence. For example, my old TI-82 can calculate and plot just about any 2D function I can imagine. My PC can simulate a world down to individual photons in real time. But if I ask my PC to make a cup of tea, it can do absolutely nothing about it without a whole lot of engineering.

Despite having the intellectual capacity of somewhere between an ant and a wasp, my PC is mathematically more competent than every human on the planet. This is because a good mathematician was able to explain complex mathematics in terms that even a computer can understand. To give an example - the average computer programmer is only expected to be able to comprehend up to 15 commands in one go.

Many of the answers given have been about problem complexity. Most complex problems can and have been broken down from being exponentially complex, to linearly complex allowing mere mortals to comprehend them (see above). For example, the Travelling Salesman Problem. In pure mathematical ability, calculating this infeasible for any human or computer. To calculate how best to travel from one end of a city to another with only 10 streets (25 intersections) would take a billion lifetimes of the Universe to calculate. Yet a satnav can resolve this problem in real time as you miss a junction.

So I would put forward Machine Intelligence. Being able to describe mathematically how to be an efficient self-learning system involves not only understanding a problem, but being able to simplify it in to a form that even a computer can calculate in linear time. See: https://en.wikipedia.org/wiki/Computational_complexity_theory

• Satnavs do not resolve the traveling salesman problem, which is believed (but not known) to be infeasible - nothing guarantees that a satnav delivers the exact most optimal route 100% of the time. – Henry Towsner Oct 26 '19 at 14:50
• Very nice suggestion, though it strays a bit from mathematics into computer science (but then, what is theoretical CS but a bastard branch of mathematics? :) ). I have some problems though with your statement that my PC is mathematically more competent than every human on the planet; conceding that this is arguing terminology, I'd say that doing mathematics is something that we haven't been able to teach computers yet: I haven't seen a PC make a real advance in math, while I've seen plenty of humans do just that. Arithmetic, however, is something that we suck at and PC's win at. – tomsmeding Oct 26 '19 at 21:27
• Photons in ray tracing are not the photons from quantum electrodynamics. So the ability of PC to do real-time ray tracing is trivial compared to the ability to simulate propagation of true photons in real time. – Ruslan Oct 26 '19 at 21:30
• Satnavs solve the "visit this one city, minimizing travel time" problem, typically using Dijkstra's algorithm. The Traveling Salesman Problem (TSP) is "visit those N cities in any order, minimizing total travel time." The "in any order" is trivial for N=1 (Dijkstra), and can be brute-forced for small N (for each permutation, perform Dijkstra for each leg), but causes the problems for large N as it quickly becomes too much to brute-force. – Sjoerd Oct 28 '19 at 22:51

Quantum cryptography algorithm proofs? Or, instead of math, how about some other branch of science or something that sounds like science to us dummies? Maybe it just has to "sound" good:

• Physics on the the event horizon of a black hole.
• Organic chemistry prior to the big bang.
• Treatment of the brain damage and psychosis caused by faster-than-light travel.
• "Organic chemistry prior to the big bang." What a novel idea. Very witty. Welcome to Worldbuilding, & have fun here. – a4android Oct 27 '19 at 22:18

One fascinating class of studies are non-associative algebras, such as loops. These are real buggers because (ab)c does not equal a(bc). That seems like a small issue, but when you are looking for an x such that a(b(c(d(e(f(gx)))))) is equal to some y, the inability to convert this to (abcdefg)x = y is a real nuisance, and its awkward:

“Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

One of the ongoing attempts to develop a Grand Unified Theory in physics is seeking to use a Moufang loop known as the octonions. It's not a fore-runner, but its in the running. Progress is slowed because so many of the tools of mathematics just don't apply in non-associative scenarios. Indeed, when I did my own searches into what loops are used for, I came up virtually empty. The traditional approach is to simply consider an associative algebra and "embed" the non-associative algebra in it, and then focus mostly on the outer associative algebra.

However, one interesting artifact that came up: knot theory. Conway famously studied knots by breaking them up into "tangles" -- knots always have the ends fuzed together like a loop, while tangles are cut open, like when one cuts a tangle out of a dog's fur. One of the interesting questions in knot theory is "are two knots equivalent," such as how a slip knot can capsize into a bowline. (it's one of the ways to make a bowline) As it turns out, the rules for manipulating these tangles form a loop -- a non-associative algebra.

Who knows. Perhaps Cat's Cradle is actually PhD level material!

I have a PhD as well and have really enjoyed reading this thread. I'm a little surprised, though, that no one has mentioned some of the basic possibilities I'm thinking of: Advanced Real Analysis; Advanced Complex Analysis; or any of the several branches of Advanced Statistical Theory, like Advanced Survival Analysis. Those are just some of the things that come to mind. Things like Fermat's last theorem and the many other things mentioned so far touch on these, but these hit the big general fields, any of which the children could study and demonstrate that they understand - to show that they are, indeed, super brilliant.

Sarah Savant's notes were a cornerstone to many women in mathematics, albeit a select few. Sarah's accomplishments in quantum economic field theory (pretend with me, here) were second to none, and her notes were sought after by many. To the regular passerby, the notes were absolutely gibberish, and there was no man who could make sense of them. They were originally published online where they could be printed in PDF form, but nobody ever made sense of those old files. It was only one day when a very special mathematician got hold of Sarah's notes. It was Sarah's daughter, Susan. Susan could make sense of these obscure notes, and over the course of a month became the most advanced mathematician in the field of quantum economic field theory. Over many years, the notes were decoded by a select few mathematicians, all of which were woman. Why?