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Elements of Geometric Algebra

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Quantum mechanics meets the dodecahedron

Yet another great image by +Greg Egan. I'll try to explain it in simple terms.

In quantum mechanics, the position of a particle is not a definite thing: it's described by a wavefunction. This says how probable it is to find the particle at any location... but it also contains other information, like how probable it is to find the particle moving at any velocity.

Take a hydrogen atom, and look at the wavefunction of the electron.

Puzzle 1. Can we make the electron's wavefunction have all the rotational symmetries of a dodecahedron - that wonderful Platonic solid with 12 pentagonal faces?

Yes! In fact it's too easy: you can make the wavefunction look like whatever you want.

So let's make the puzzle harder. Like everything else in quantum mechanics, angular momentum can be uncertain. In fact you can never make all 3 components of angular momentum take definite values! However, there are lots of wavefunctions where the magnitude of the angular momentum is completely definite.

Puzzle 2. Can an electron's wavefunction have a definite magnitude for its angular momentum while having all the rotational symmetries of a dodecahedron?

Yes! And there are infinitely many ways for this to happen! +Greg Egan drew the simplest one here:

and this started a long discussion. By the end, we had completely crushed the problem. So, we could solve harder puzzles.

The magnitude of the angular momentum is determined by a number called ℓ, for some idiotic reason. And this number is quantized! It can only take values 0, 1, 2, 3, ... and so on.

The simplest solution to Puzzle 2 has ℓ = 6, for some reason that's not at all idiotic. We can get it using the 6 lines connecting opposite faces of the dodecahedron!

How does that work? Well, read the discussion on Egan's post. It takes some math to see how it works.

Puzzle 3. What's the smallest choice of ℓ where we can find two different electron wavefunctions that both have the same ℓ and both have all the rotational symmetries of a dodecahedron?

It turns out to be ℓ = 30. The picture on this post shows a wavefunction oscillating between these two possibilities!

But we can go a lot further:

Puzzle 4. For each ℓ, how many linearly independent electron wavefunctions have that value of ℓ and all the rotational symmetries of a dodecahedron?

For ℓ ranging from 0 to 29 there are either none or one. There are none for these numbers:

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29

and one for these numbers:

0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28

(I said the case ℓ = 6 gave the simplest wavefunction with dodecahedral symmetry, but I was lying. The case ℓ = 0 gives a constant wavefunction. This has dodecahedral symmetry, but it's completely boring: the picture would be a featureless sphere!)

The pattern continues as follows. For ℓ ranging from 30 to 59 there are either one or two. There is one for these numbers:

31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 47, 49, 53, 59

and two for these numbers:

30, 36, 40, 42, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58

The numbers in these lists are just 30 more than the numbers in the first two lists! And it continues on like this forever.

Puzzle 5. What's special about these numbers from 0 to 29?

0, 6, 10, 12, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28

You don't need to know tons of math to figure this out - but I guess it's a sort of weird pattern-recognition puzzle unless you know the math that says which patterns are likely to be important here. So, as a hint, I'll say that writing numbers as sums of other numbers is important.

Animated Photo

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Scientists made use of mathematical models of algebraic topology in order to describe different structures and multidimensional geometric spaces in human brain networks.

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In algebraic topology you work with something more general than Euler Riemann surfaces: Simplical complexes! Instead of the Euler characteristic you use Betti numbers as an invariant to also classify higher dimensional surfaces.

If you're up to learn more about algebraic topology you could warm up with the latest Infinite Series video here:

And then get on and read Allen Hatcher's excellent free book called "Algebraic Topology" here:

If you're not convinced or satisfied yet, try tracing the trail that lead me to write this post:

Recently, +John Baez gave a talk about 'how algebraic topology is changing our view of mathematical reality', you can find the slides, some of the crowd and a comment not quite unlike this post here:

And here are some trails on the n-cat cafe:

#algebraic #topology

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How math is changing

People are starting to use algebraic topology — basically the art of counting holes of different kinds — to study patterns in data. It's called topological data analysis.

I just learned a lot about this at a conference called "Applied Algebraic Topology 2017". But my own talk was not about applications. Instead, it was about how algebraic topology is changing our view of mathematical reality!

What's happening is that sets are slowly becoming less important, and spaces are becoming more fundamental. In the process, the concept of 'space' is getting more flexible.

So far this is mainly happening in the most sophisticated branches of pure math. You might not notice if you're working down in the trenches. But I think topological data analysis is a sign that this trend is spreading. We can now describe complicated spaces with interesting holes using a finite amount of data — suitable for computing.

Where will it lead? Nobody knows! You can see the story so far in my talk slides. But they're pretty hard-hitting, since I was talking to folks who know algebraic topology. So it might be better to hear what Yuri Manin said when he was asked what the future holds.

He started by saying "I don’t foresee anything extraordinary in the next twenty years." But then he described something pretty extraordinary:

...after Cantor and Bourbaki, no matter what we say, set theoretic mathematics resides in our brains. When I first start talking about something, I explain it in terms of Bourbaki-like structures: topological spaces, linear spaces, the field of real numbers, finite algebraic extensions, fundamental groups. I cannot do otherwise. If I’m thinking of something completely new, I say that it is a set with such-and-such a structure; there was one like this before, called this-and-that; another similar one was called this-and-this; so I apply slightly different axioms, and I will call it such-and-such. When you start talking, you start with this. That is, at first we start with the discrete sets of Cantor, upon which we impose something more in the style of Bourbaki.

But fundamental psychological changes also occur. Nowadays these changes take the form of complicated theories and theorems, through which it turns out that the place of old forms and structures, for example, the natural numbers, is taken by some geometric, right-brain objects. Instead of sets, clouds of discrete elements, we envisage some sorts of vague spaces, which can be very severely deformed, mapped one to another, and all the while the specific space is not important, but only the space up to deformation. If we really want to return to discrete objects, we see continuous components, the pieces whose form or even dimension does not matter. Earlier, all these spaces were thought of as Cantor sets with topology, their maps were Cantor maps, some of them were homotopies that should have been factored out, and so on.

I am pretty strongly convinced that there is an ongoing reversal in the collective consciousness of mathematicians: the right hemispherical and homotopical picture of the world becomes the basic intuition, and if you want to get a discrete set, then you pass to the set of connected components of a space defined only up to homotopy.

That is, the Cantor points become continuous components, or attractors, and so on — almost from the start. Cantor’s problems of the infinite recede to the background: from the very start, our images are so infinite that if you want to make something finite out of them, you must divide them by another

Here's the abstract of my talk. This should either make you curious enough to look at the slides, or confused enough to be happy you didn't:

Abstract. As algebraic topology becomes more important in applied mathematics it is worth looking back to see how this subject has changed our outlook on mathematics in general. When Noether moved from working with Betti numbers to homology groups, she forced a new outlook on topological invariants: namely, they are often functors, with two invariants counting as "the same" if they are naturally isomorphic. To formalize this it was necessary to invent categories, and to formalize the analogy between natural isomorphisms between functors and homotopies between maps it was necessary to invent 2-categories. These are just the first steps in the "homotopification" of mathematics, a trend in which algebra more and more comes to resemble topology, and ultimately abstract "spaces" (for example, homotopy types) are considered as fundamental as sets. It is natural to wonder whether topological data analysis is a step in the spread of these ideas into applied mathematics, and how the importance of "robustness" in applications will influence algebraic topology.

The slides are here:


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The main goal of my exposition shall be to

• describe how schemes are a generalization of varieties
• or, more precisely, how varieties are a special case of schemes,
• or, more precisely, how the category of varieties is a subcategory of that of schemes,
• or, to be really precise, how there is a fully faithful functor

from the category of varieties over a field k, into that of schemes over a field k.

In order to be even able to state this result, I will need to say a few words about what varieties and schemes actually are.

Doing so already goes a long way towards proving the above statement, since the definition of a scheme is precisely motivated by the desire to generalize that of a variety.

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