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

Question

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.

Answer 1

Main Answer

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.

Comments

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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

Answer 6

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.

Answer 7

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.

Answer 8

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.

Here are useful links to

• a Mathoverflow soft question on modern algebraic geometry

• The Wikipedia article on the foundational treatise

Answer 9

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:

• The Riemann hypothesis, which has been open for 160 years (since 1859), and would probably be the consensus choice among mathematicians for the most important open problem in mathematics.

• The P vs. NP problem, which quite important in both mathematics and computer science and probably the second most famous of the Millennium Problems.

• The Birch and Swinnerton-Dyer conjecture. Solving this would also solve the congruent number problem, which has been open since 972 AD.

• The existence of a Yang-Mills theories with positive mass gap, which has implications in both mathematics and quantum physics.

• The Hodge conjecture

• The existence of solutions to the Navier-Stokes equation

Other long-standing open problems include:

• The odd perfect number problem, which dates to ancient Greek times (100 AD or so) and is probably the oldest open problem in mathematics.

• The twin prime conjecture. (Note: There has been significant progress on this lately, and it wouldn't be too surprising for it to be solved in a decade or so, meaning that it might not be the best choice.)

• Goldbach's conjecture.

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.

Answer 10

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.

Answer 11

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.

Watch this and see if it answers any of your questions.

Answer 12

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.

Answer 13

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).

Answer 14

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

Answer 15

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.

Answer 16

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!

Answer 17

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.

Answer 18

I'd somewhat dislike real-world mathematics portrayed in the way you're looking for, because I, personally, care about maths accessibility and would like to uphold that many if not most people can learn to do even the most advanced subjects. There's an ongoing debate whether you need to be a genius to do mathematics. There are also two cultures in mathematics, of 'problem solvers' and 'theory builders'. Some areas of mathematics have easily stated problems that require little background, yet have very complex solutions that may perhaps require more of a stroke of 'genius'. Other areas are more scholarly in nature, and you need to immerse yourself in extensive literature and conceptual background. Also in the 21st century, mathematics like all science is increasingly becoming a collective effort rather than an individual one. An important part of mathematical progress is thus making it accessible to others. (None of this is to diminish the enormous impact that outstanding individual thinkers have had on various fields)

Now for the purpose of your fiction, it also needs to be credible why the society chooses to impose a significant risk on its children just to advance area X of mathematics, as important as it may be. I would instead begin from the purpose, and then find the required field of excellence to fit it.

The idea of enhancement for some purpose is a common and interesting one in fiction. I'd say that technology is probably a better driver for such things than pure theory. So let me give some ideas (sorry if they don't answer the question directly).

• Particle physics. Say breakthroughs towards a Grand Unified Theory have been made, but the maths is ridiculously complicated. Nations/planets are in an arms race situation, and you need to have the next Einstein quickly.
• Quantum computation. With the widespread use of quantum computation, being able to visualize configurations of qubits in high-dimension space becomes an important advantage. We have the Bloch sphere for 1 qubit, but for n>1 visualizing all possible axes of correlation/entanglement becomes extremely hard.
• Navigation in 4d space. Think Dune's navigators. Generally any cognitive task arising from the neural connection of humans to machines
• Strategy. Think Ender's Game.

Answer 19

Generally the human brain can not deal intuitively with calculation in more than three geometric dimensions. As a result, calculations in multi-dimensional spaces have been used as an indicator of a very advanced mind in science fiction for years. It is near cliché.

On a side note, remember that you have to make the whole thing easy for non mathematically minded people to understand.

Answer 20

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?

The answer to both questions:

Sarah, and Susan were both tetrachromads, and the notes only made sense when 4 base colors were taken into account