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What is a manifold?

quantamagazine.org

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Re: What is a manifold?

#51

Post by tamnd »

lizknope wrote: post #43

I was reading a book on string theory and I remember the Calabi–Yau manifold https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold I'm not going to pretend to understand it all but they do make pretty pictures! https://www.google.com/search?q=calabi+yau+manifold+images

I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing.

What makes Calabi Yau manifolds special is that their curvature balances out perfectly so the space does not stretch or shrink overall.

In physics especially in string theory Calabi Yau manifolds are used to describe extra hidden dimensions of the universe beyond the three we can see. The shape of a Calabi Yau manifold affects how particles and forces behave which is why both mathematicians and physicists study them.

Re: What is a manifold?

#52

Post by mmooss »

thaumasiotes wrote: post #47

> They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi (opens a new tab), a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.” Something's gone badly wrong here. "Without learning Cyrillic" is the normal way to learn Russian. Pick a slightly less prominent language and 100% of learn…

I thought the same - many languages don't have a writing system and children learn without being able to write. But that's beside the point; the point is just as valid even if the analogy is poor.

Re: What is a manifold?

#53

Post by tamnd »

WhyOhWhyQ wrote: post #49
tamnd wrote: post #48

I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.

It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.

That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation.

I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.

Re: What is a manifold?

#54

Post by mulmen »

aleph_minus_one wrote: post #38
hufdr wrote: post #29

Earlier quoted context omitted.

Quanta’s greatest strength is that it doesn’t pretend to be clever. Many tech publications write as if they’re showing off, and you just end up feeling tired after reading them.

> 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.

Seems superficial. If a simple concept is presented in a complex way what did you actually learn?

Re: What is a manifold?

#55

Post by aleph_minus_one »

mulmen wrote: post #54
aleph_minus_one wrote: post #38

Earlier quoted context omitted.

> 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.

Seems superficial. If a simple concept is presented in a complex way what did you actually learn?

Often, if the concept is presented in a more complex way the reason is that the author wants to emphasize and explain how the concept relates in a non-trivial way to some other deep concept; thus you learn a lot more than when the author explains things in the most simple (and shallow) way.

Re: What is a manifold?

#56

Post by senderista »

omnicognate wrote: post #8
elashri wrote: post #6

This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers. Looking at things from abstract view does allow us not to worry about how we visual…

This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations. In special relativity, for example, a…

One of the worst examples is Weinberg’s book on GR, which I found nearly unreadable due to the morass of coordinates/indices. So much more painful to learn from than Wald or other mathematically modern treatments of GR.

Re: What is a manifold?

#58

Post by moralestapia »

tamnd wrote: post #51
lizknope wrote: post #43

I was reading a book on string theory and I remember the Calabi–Yau manifold https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold I'm not going to pretend to understand it all but they do make pretty pictures! https://www.google.com/search?q=calabi+yau+manifold+images

I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing. What makes Calabi Yau manifolds spe…

>their curvature balances out perfectly so the space does not stretch or shrink overall

Could you elaborate a bit on this? I find it fascinating. Thanks.

>The shape of a Calabi Yau manifold affects how particles and forces behave [...]

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

Re: What is a manifold?

#59

Post by brosco »

mmooss wrote: post #50

I rarely see manifolds applied directly to cartographic map projections, which I've read about a bit, though the latter seem like just one instance of the former. Does anyone know why cartographers don't use manifolds, or mathematicians don't apply them to cartography? (Have I just overlooked it?)

One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already.

Another is that when working with manifolds, you usually don't get a set of global coordinates. Manifolds are defined by various local coordinate charts. A smooth manifold just means that you can change coordinates in a smooth (differentiable) way, but that doesn't mean two people on opposite sides of the manifold will agree on their coordinate system. On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.

I'm not very well versed in the history, but the study of cartography certainly predates the modern idea of an abstract manifold. In fact, the modern view was born in an effort to unify a lot of classical ideas from the study of calculus on spheres etc.

Re: What is a manifold?

#60

Post by senderista »

omnicognate wrote: post #8
elashri wrote: post #6

This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers. Looking at things from abstract view does allow us not to worry about how we visual…

This is a tendency among physicists that I find a bit painful when reading their explanations: focusing on how things transform between coordinate systems rather than on the coordinate-independent things that are described by those coordinates. I get that these transformation properties are important for doing actual calculations, but I think they tend to obfuscate explanations. In special relativity, for example, a…

I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.
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