Lee taught Intro to Topological Manifolds for one quarter, and then the next two quarters where Intro to Smooth Manifolds. Then Riemannian, then vector bundles, and then complex manifolds.

Oh dope. Added to my feedbin!

Remarkably, they don't even ask for money anywhere on the site. Now that is a rare thing on the modern internet, especially for high quality writing.

> Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.

I like this honestly because this shows that I learned something intelligent. On the other hand, if I don't feel exhausted after reading, it is a strong sign that the article was below my intellectual capacity, i.e. I would have loved it if I could have learned more.

Carnegie Libraries, Nobel Prizes, Rhodes scholarships?

> Could you elaborate a bit on this?

Please correct me if I am wrong, I have not touched this subject in a long time and only have some intuition. Here is how I understand it:

A manifold is a kind of space that looks flat when you zoom in close enough. The surface of a sphere or a doughnut is a 2D manifold, and the space we live in is a 3D manifold. A Calabi Yau is one of these spaces but with more dimensions and extra symmetry that makes it very special.

In geometry there are several ways to describe curvature. The most complete one is the Riemann curvature tensor, which contains all the information about how space bends. If you take a specific kind of average of that, you get the Ricci curvature tensor. Ricci curvature tells you how the size of small regions in space changes compared to what would happen in flat space.

Imagine a tiny ball floating in this curved space. If the Ricci curvature is positive, nearby paths tend to come together and the ball’s volume becomes smaller than it would in flat space. If the Ricci curvature is negative, nearby paths move apart and the ball’s volume grows larger. If the Ricci curvature is zero, the ball keeps the same volume overall. So when I said “the space does not stretch or shrink overall” I was describing this situation: the Ricci curvature is zero, which means the space does not expand or contract on average compared to flat space.

The space can still have complicated twists and bends. Ricci curvature only measures a certain type of curvature related to volume change. Even if the Ricci tensor is zero, there can still be other kinds of curvature present. The curvature balances out is just an intuitive way to express that the volume effects cancel when you take the average that defines Ricci curvature. It does not mean the space has matching regions of positive and negative curvature in a literal sense, but rather that the mathematical combination producing Ricci curvature sums to zero.

Noe back to definition: A Calabi-Yau manifold is defined as compact (finite in size), complex), and Kähler (it has a compatible geometric and complex structure), with a first Chern class equal to zero. Yau’s theorem proves that such a space always has a way to measure distances so that its Ricci curvature is exactly zero. So when I said “the curvature balances out perfectly so the space does not stretch or shrink overall” I meant it as an intuitive description of this Ricci flat property. The space is not flat like a sheet of paper, but its internal geometry is perfectly balanced in the sense that there is no net expansion or contraction of space.

> Do you know if there's any experimental evidence of this?

As my knowledge, there is no direct evidence that Calabi Yau manifolds describe real extra dimensions. In string theory, these shapes are used because they fit the math and preserve symmetries like supersymmetry. Experiments have not found signs of extra dimensions or supersymmetric particles, so Calabi Yau manifolds remain a beautiful theoretical idea, not something confirmed by observation.

That's good to know about Wald. I bought a copy to finally get my head round General Relativity, but its brief explanation of Special Relativity right at the start made it clear that I hadn't properly understood that, which led to me getting Gourgoulhon's book. I should be better placed to tackle it now.

> The issue is the level of mathematical sophistication one has when a certain concept is introduced. That often defines or at least heavily influences how one thinks about it forever.

That was one of the most interesting things of my EE/CS dual-degree and the exact concept you're describing has stuck with me for a very long time... and very much influences how I teach things when I'm in that role.

EE taught basic linear algebra in 1st year as a necessity. We didn't understand how or why anything worked, we were just taught how to turn the crank and get answers out. Eigenvectors, determinants, Gauss-Jordan elimination, Cramer's rule, etc. weren't taught with any kind of theoretical underpinnings. My CS degree required me to take an upper years linear algebra course from the math department; after taking that, my EE skills improved dramatically.

CS taught algorithms early and often. EE didn't really touch on them at all, except when a specific one was needed to solve a specific problem. I remember sitting in a 4th year Digital Communications course where we were learning about Viterbi decoders. The professor was having a hard time explaining it by drawing a lattice and showing how you do the computations, the students were completely lost. My friend and I were looking at what was going on and both had this lightbulb moment at the same time. "Oh, this is just a dynamic programming problem."

EE taught us way more calculus than CS did. In a CS systems modelling course we were learning about continuous-time and discrete-time state-space models. Most of the students were having a super hard time with dx/dt = A*x (x as a real vector, A as a matrix)... which makes sense since they'd only ever done single-variable calculus. The prof taught some specific technique that applied to a specific form of the problem and that was enough for students to be able to turn the crank, but no one understood why it worked.

Yeah, I had a slightly odd introduction to these things as I studied joint honours maths and physics. That meant both that I had a bit more mathematical maturity than most of the physics students and that I was being taught the more rigorous underpinnings of the maths while it was being (ab)used in all sorts of cavalier ways in physics. I liked the subject matter of physics more, but I greatly preferred the intellectual rigour of the maths.

Eric Gourgoulhon is a product of the French education system, and I often think I would have done better studying there than in the UK.

> a convenient shorthand for the bit that I put in angle brackets above.

Yes, but the "convenient shorthand" only makes sense if you already know what a tensor is. That renders the "definition" useless as an explanation or as pedagogy. It's only useful as a social signal to let others know that you understand what a tensor is (or at least you think you do).

> My favourite explanation is that "Tensors are the facts of the universe"

That's not much better. "The earth revolves around the sun" is a fact of the universe, but that doesn't help me understand what a tensor is.

What matters about tensors are the properties that distinguish them from other mathematical objects, and in particular, what distinguishes them from closely related mathematical objects like vectors and arrays. Finding a cogent description of that on the internet is nearly impossible.

> the reality of the tensor ... is independent of the coordinate system chosen by the observer

Now you're getting closer, but this still misses the mark. What is "the reality of a tensor"? Tensors are mathematical objects. They don't have "reality" any more than numbers do.

> no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones

That is closer still. But I would go with something more like: tensors are a way to represent vectors so that the representation of a given vector is the same no matter what basis (or coordinate system) you choose for your vector space.

The problem is that this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). Manifolds are required to have an "atlas"[0]: a collection of coordinate systems ("charts") that cover the space and are continuous mappings from open subsets of the underlying topological space to open subsets of Euclidean space, with the overlaps between charts inducing smooth (i.e., infinitely differentiable) mappings in Euclidean space.

Colloquially, this means a manifold is just "a bunch of patches of n-dimensional Euclidean space, smoothly sewn together."

A sphere requires at least two charts for an admissible atlas (say two hemispheres overlapping slightly at the equator, or six hemispheres with no overlaps), otherwise you get discontinuities.

[0] https://en.wikipedia.org/wiki/Atlas_(topology)