(With apologies and tribute to the late Adam West)
Here’s a situation known to any beginning skier.
You are at the top of a mountain slope. You want to go down to the bottom of the slope. But you are on skis, and you are not very good at skiing.
Despite your lack of skill, you have been deposited by a ski lift at the top of the slope. Slightly terrified, you know that the only honourable way out of your predicament is down — down the slope, on your skis.
Pointing your skis down the slope, with a rush of exhilaration you find yourself accelerating downwards. Unfortunately, you know from bitter experience that the hardest thing about learning to ski is learning to control your speed.
If you are bold or stupid, your incompetent attempt to conquer the mountain likely ends in a spectacular crash. If you are cowardly and have mastered the snowplough, your plodding descent ends with a whimper and worn out thighs from holding the skis in an awkward triangle all the way down.
But you know that the more your skis point down the mountain, the faster you go. And the more your skis point across the slope, the slower you go.
When your skis point down the slope, they are pointing steeply downwards; they are pointing in quite a steep direction. But when your skis point across the slope, they are not pointing very downwards at all; they are pointing in quite a flat direction.
As an incompetent skier, the best way to get down the slope without injury and without embarrassment is to go take a path which criss-crosses the slope as much as possible. You want your skis to point in a flat direction as much as possible, and in a steep direction as little as possible.
The problem is that each time you change direction, you temporarily point downwards, and risk runaway acceleration.
Is it possible to get down the mountain while always pointing in a flat direction?
Of course the answer is no. But there is an area of mathematics which says that the answer is yes, almost, more or less.
This, very roughly, is the beginning of the idea of Gromov’s homotopy principle — often abbreviated to h-principle.
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The ideas of the h-principle were developed by Mikhail Gromov in his work in the 1970s, including work with my PhD advisor Yakov Eliahsberg. The term “h-principle” first appeared and was systematically elaborated by Gromov in his (notoriously difficult) book Partial differential relations. Gromov, who grew up in Soviet Russia, was not a friend of the authorities and in 1970, despite being invited to speak at the International Congress of Mathematicians in France, was prohibited from leaving the USSR. Later he finally made his “gromomorphism” to the US, and now he works at IHES just near Paris.
The ideas of the h-principle are about a “soft” kind of geometry. If you are prepared to treat your geometry like playdough, and morph various things around, within certain constraints, then the $h$-principle tells you how to morph your playdough-geometry to get the nice kind of geometry you want. Technically, morphing playdough has a fancy name: it’s called homotopy.
An example of the h-principle (or, more precisely, of “holonomic approximation”) in the skiing context would be as follows.
Your ideal skiing path down the slope would be to go straight down, but always have your skis pointing flat. That is, your skis should always point horizontally, but you should go straight down the mountain. That is the ideal.
That, of course, is a ridiculous ideal path. But it’s an ideal path nonetheless. You want to go straight down the mountain, and the ideal path does this; and you want to have your skis pointing safely horizontally, and the ideal path does this too. This is the ideal of the incompetent skier: go down the mountain as directly as possible, with your skis always being completely flat.
Now the ideal path cannot be achieved in practice. In practice, if your skis are pointing horizontally, you go horizontally. (We ignore skidding for the purposes of our mathematical idealisation.) In practice, you go in the direction your skis are pointing.
The “ideal path” is a path down the mountain, which also tells you which way your skis should point at each stage (i.e. horizontally), but which doesn’t satisfy the practical principle that you should go in the direction you’re pointing. As such, it’s a generalisation of the real sort of path you could actually take down the mountain. (The technical name for this type of path is a “section of a jet bundle”.)
A realistic path down the mountain — one where you go in the direction your skis are pointing — is also known as a holonomic path.
One of the first results towards the h-principle, says that if you are prepared to make a few tiny tiny adjustments to your path, then you can take an actual, holonomic path down the mountain, where you are very very close to the ideal path — both in terms of where you go, and in the direction your skis point. You stay very very close — in fact, arbitrarily close — to the path straight down the mountain. And your skis stay very very close — again, arbitrarily close — to horizontal at every instant on the way down.
How is this possible? Well, you have to make some adjustments.
First, you make some adjustments to the path. You might have to make a wiggly path, rather than going in a straight line. Actually, it will have to be very wiggly — arbitrarily wiggly.
And, second, you’ll have to make some adjustments to the mountain too. You’ll have to adjust the shape of the mountain slope — but only by a very very small, arbitrarily small amount.
Well, perhaps these types of alternations are rather drastic. But without moving the mountain, you won’t be able to go down the mountain and stay very close to horizontal. You must alter the ski slope, and you must alter your path. But these movements are very very small, and you can make them as small as you like.
How do you alter the mountain? Roughly, you can make tiny ripples in the slope — and roughly, you turn it into a terraced slope. Just like rice farming in Vietnam, or for growing crops in the Andes.
As you go along a terrace, you remain horizontal! We don’t want our terraces to be completely horizontal though — we want them to have a very gentle downwards slope, so that we can stay very close to horizontal, and yet eventually get to the bottom of the mountain.
And also, we’ll need to be able to go smoothly from one terrace down to the next, so each terraces should turn into the next. So perhaps it’s more like Lombard Street, the famously windy street in San Francisco. (Which is not, however, the most crooked street in the US — that’s Wall Street, of course. Got you there.)
Perhaps a more accurate depiction of what we want is the figure below, from Eliashberg and Mishachev’s book Introduction to the h-principle. We want very fine, very gently sloping terraces, and we want them extremely small, so that the mountain is altered by a tiny tiny amount. And to go down the slope we need to take a very windy path — with many many wiggles in it. To go down the slope is almost like a maze — although it’s a very simple, repetitive maze.
Thus, with a astronomical number of microscopic terraces of the mountain, each nano-scopically sloped downwards, and an astronomical number of microscopic wiggles in your path down the terraces, you can go down the mountain, staying very close to your idealised path. You go very very close to straight down the mountain, and your skis stay very very close to horizontal all the way down.
And then, you’ve done it.
This is the flavour of the h-principle. More precisely this result is called holonomic approximation. Holonomic approximation says that even an incompetent skier can get down a mountain with an arbitrarily small amount of trouble — provided that they can do an arbitrarily large amount of work in advance to terrace the mountain and prepare themselves an arbitrarily flat arbitrarily wiggly path.
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The h-principle has applications beyond idealised incompetent skiing down microscopically terraced mountains. Two of the most spectacular applications are sphere eversion, and isometric embedding. In fact they both preceded the h-principle — and Gromov’s attempt to understand and formalise them directly inspired the development of the h-principle.
Sphere eversion is a statement about spheres in three-dimensional space. Take a standard unit sphere in, but again regard it as made of playdough, and we will consider morphing (erm, homotoping) it in space. We allow the sphere to pass through itself, but never to crease, bend or rip. All the sphere can intersect itself, each point of the sphere must remain smooth enough to have a tangent plane. (The technical name for this is an immersion of the sphere into 3-dimensional space.)
Smale’s sphere eversion says that it’s possible to turn the sphere inside out by these rules — that is, by a homotopy through immersions. This amazing theorem is all the more amazing because Smale’s original 1957 proof was an existence proof: he proved that there existed a way to turn the sphere inside out, but did not say how to do it! Many explicit descriptions have now been given for sphere eversions, and there are many excellent videos about it, including Outside In made in the 1990s. My colleague Burkard Polster, aka the Mathologer, also has an excellent video about it.
Smale has an interesting and admirable biography. He was actively involved in the anti-Vietnam War movement, even to the extent of being subpoenaed by the House Un-American Activities Committee. His understanding of the relationship between creativity, leisure and mathematical research was epitomised in his statement that his best work was done “on the beaches of Rio”. (He discovered his horseshoe map on the beach at Leme.)
Isometric embedding is more abstract; see my article in the conversation for another description, but it is even more amazing. It is a theorem about abstract spaces. For instance, you could take a surface — but then, put on it an abstractly-defined metric, unrelated to how it sits in space. Isometric embedding attempts to map the surface back into 3-dimensional space in a way that preserves distances, so that the abstract metric on a surface corresponds to the familiar notion of distance we know in three dimensions.
Isometric embedding is largely associated with John Nash, who passed away a couple of years ago and is more well known for his work on game theory, and from the book and movie A Beautiful Mind. The proof is incredible. Gromov describes how he came to terms with this proof in some recollections. He originally found Nash’s proof “as convincing as lifting oneself by the hair”, but after eventually finding understanding, he found Nash’s proof “miraculously, did lift you in the air by the hair”!
The Nash-Kuiper theorem says that if you can map your abstract surface into 3-dimensional space in such a way that it decreases distances, then you can morph it — homotope it — to make it preserve distances. (Actually, it need not be a surface but a space of any dimension; and it need not be 3-dimensional space, but space of any dimension.) And, just like on the ski slope, this alteration of the surface in 3-dimensional space can be made very very small — arbitrarily small.
The h-principle is another mathematical superpower, and it comes up in many places where geometry is “soft”, and you can slightly “morph” or “adjust” your geometrical situation to find the situation we want.