quantum-integration.org

There are several very generic mathematical abstractions, which I miss a lot in my daily life. The issue is that those concepts do exist, and I meet them all the time, but I do not know how to refer to them without using a mathematical term, that most people will not understand.

So, to celebrate the Pi Day, let's try to close the gap.

And today we are going to talk about the idea of a Manifold.

The elephant in the room

The Wikipedia article for manifold shares the same problem as many other mathematical pages – it dives directly into the technical details without trying to explain the idea in “layman’s terms”, which makes it not accessible for anyone without a math degree.

So let’s take another path and start by recalling the Parable of Blind Men and an Elephant:

“a story of a group of blind men who have never come across an elephant before and who learn and imagine what the elephant is like by touching it. Each blind man feels a different part of the animal's body, but only one part, such as the side or the tusk. They then describe the animal based on their limited experience and their descriptions of the elephant are different from each other”

What if I tell you that topological manifold is a representation of this very idea?

Since I don’t like torturing neither people nor animals for the sake of an imaginary argument, let’s change a setting a bit and assume that we have a large rock in a dark room. We have a small torchlight, so at any moment in time we can observe a single small spot on that rock.

And the big question of topology is – how much I can figure out about the whole object, given the knowledge of small neighborhoods of every single point.

The ants

Now the torchlight analogy is actually cheating. If you are running around the object with a torchlight, you already have quite a lot of information. You can step back and take a better look, or you can measure distance between any two points with a ruler. The real fun starts when you can not look at the object from the safety and comfort of a familiar (often Euclidean) embedding space.

So rather than being a person observing the rock from the side, imagine to be the ant living on it. You don’t have a universal ruler, you are living on your small area of the rock, tracing your own tiny steps and measuring them with your own tiny feet. The thick fog covers everything, so that you can not look at the distance, therefore you live in the down-to-earth two-dimensional world.

And then there are other ants, living in their own areas.

And in the evening all those ants from all over the place join their federated social network and discuss the topics like “is the Earth flat?” and “what is the shortest path from the Green Moss Valley to the Hard Rock Mountain?”.

To prove the point they draw charts of the area, each their own. But then they quickly realize that they do not really know how to compare the foot of a red ant with a foot of the green one, and whether the Green Moss Valley one ant has drawn on their chart is the same Green Valley as drawn by the other.

Because creating the collection of local charts is not enough.

I mean, do not get me wrong, it is definitely better than nothing. At least you can confirm that you are ants living on a two dimensional surface, and you are not talking with a jellyfish floating in a three dimensional liquid. But that’s basically all you can get.

This stage of our ant community is called a topological manifold (“n-manifold is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space”).

The atlas

The use of the word “atlas” in a geographical context dates from 1595 when the German-Flemish geographer Gerardus Mercator published Atlas Sive Cosmographicae Meditationes de Fabrica Mundi et Fabricati Figura (“Atlas or cosmographical meditations upon the creation of the universe and the universe as created”). This title provides Mercator's definition of the word as a description of the creation and form of the whole universe, not simply as a collection of maps.

Wikipedia, etymology of the word “atlas”.

The simple collection of charts doesn’t give you any idea about how the charts relate to each other. We are in that blind elephant situation, where I can talk about my map, you can talk about yours, but there is no connection between them. And we don’t have a third-party, the embedding space, which would provide us a shared frame of reference.

How do we fix it? We create the connection.

Every ant needs to reach their neighbors – those other ants whose area overlaps with this one. And for that overlapping part they have to create a mapping between their respective charts – to calculate how many red feet are in a green foot, and to pin the shared known locations.

It is important that all the charts are compatible, so that those mappings, transitional functions between every pair of overlapping charts, are well-defined. This is not a given. There are, in fact, topological manifolds which are covered by charts in such a way that there is no good mapping between them.

If our ant community manages to achieve that level of agreement, they are upgraded from the status of a topological manifold to a differentiable or a smooth one.

The happy end

And this is the moment when we reach some shared level of understanding:

if one ant wants to compare notes with a remote ant somewhere far away, they first establish a path via chain of neighborhoods. And then those transition functions allow them to carry notions along the path.

What I like about the manifold object, as opposed to the elephant parable, is that rather than just talking about a problem of misunderstanding it gives you a recipe to resolve it:

Find the overlapping areas, create the mappings, ensure these are compatible, connect your local charts with the path, and walk it step by step applying transitions, when required.

There is more to it, of course. Do we always get the path? Do we get the same result if we choose another path to the same destination? How ants identify valleys and mountains? Can a donut-shaped Earth still be flat?..

This is what Geometry and Topology is about.

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And every so often when I see an argument, I want to ask: folks, have you checked your transition functions? :)