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.
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.
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.
Tbh, I never quite understood the appeal of John M. Lee's book. It's not bad but I didn't find it great, either, especially (IIRC) in terms of rigor. Meanwhile, the much less well-known "Manifolds and Differential Geometry" by Jeffrey M. Lee (yeah, almost the same name) was much better.
This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.
It’s actually much more well written than the majority or articles we usually come across.
And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/
Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain.
For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as a square [O,2π[²?
I also expected more talk about atlases. In simple cases, it's easy to think of a surface as a deformation of a flat shape, so a natural idea is to think of having a map from the plan to the surface. But, even for a simple sphere, most surfaces can't map to a single flat part of the plan, and you need several maps. But how do you handle the parts where the maps overlap? What Riemmann did was defining properties on this relationship between manifold points and maps (which can be countless).
BTW, I know just enough about relativity to deny that "space-time [is] a four-dimensional manifold", at least a Riemmannian manifold. IIRC, the usual term is Minkowski-spacetime.
> Precisely, why more as a manifold than as a square
In a double pendulum, each arm can freely rotate (there is no stopping point). This means 0 degrees and 360 degrees are the same point, so the edges of the square are actually joined. If you join the left and right edges to each other, then join the top and bottom edges to each other, you end up with a torus.
Minkowski spacetime is the term in special relativity, i.e. the flat case, or zero curvature. In general relativity, spacetime is a pseudo Riemannian manifold, like the sibling comment says. Unlike Minkowski spacetime, it can be curved.
Spacetime is a four-dimensional manifold (at least theoretically - who knows what it is in reality). Technically it's a pseudo-Riemannian manifold since the metric is not positive definite: it can be negative or zero for non-zero vectors. A Riemannian manifold proper has a positive definite metric, but in popularizations like this I wouldn't really expect them to get into these kinds of distinctions.
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles.
Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either
Quanta is amazing because it doesn't have to worry about money. It's a publication run by the Simons Foundation, funded with the proceeds of the wildly successful RenTec hedge fund. So they get pretty much full editorial control.
For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.
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.
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.
Also speaks to a lack of understanding on the author's part; people who truly understand some subject are generally much more adept at explaining it in simpler terms – ie without adding complexity beyond the subject's essential complexity
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Good game and a hard question, especially if you make "comparable" more explicit. I'd add "noncommercial, open-access", and "modern" in the sense that it happened under the current norms with respect to legacy and the social contract.
I agree. I find their articles very enjoyable. And even though they stay technical, they don’t descend into becoming a technical journal. The content is still accessible to a non-expert like me.
Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.
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 visualize the geometry which is actually hard and sometimes counter intuitive.
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 huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.
For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.
I agree that focusing on Lorentz transformations is the wrong way to approach thinking about special relativity. But It might be the right way to teach it to physics students.
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.
The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.
The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".
(As an aside: I really love abstract index notation for dealing with tensors)
> 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.
> But It might be the right way to teach it to physics students.
Having studied physics, I would disagree rather strongly. I only really started understanding Special Relativity once I had a clear understanding of the math. (And then it becomes almost trivial.) Those of my fellow class mates, however, who didn't take the time to take those additional (completely optional) math classes, ended up not really understanding it at all. They still got confused by what it all meant, by the different paradoxes, etc.
I saw the same effect when, later, I was a teaching assistant for a General Relativity class.
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.
I had started in a theoretical physics degree which was jointly taught by the maths and physics department. By my final year I had changed into an ostensibly pure maths degree, although I did it mainly to take more advanced theoretical/mathematical physics courses (which were taught by the maths department), and avoid having to do any lab work—a torsion pendulum experiment was my final straw on that one, I don't know what caused it to fuck up, but fuck that.
In the end I took on more TP courses than the TP students, nearly burnt out by the end of the year, and... didn't exactly come out with the best exam results.
Taylor & Wheeler's Spacetime Physics is similar. They emphasize the importance of frame invariant representations. (I highly recommend the first edition over the second edition, the second edition was a massive downgrade.)
Kip Thorne was also heavily influenced by this geometric approach. Modern Classical Physics by Thorne & Blandford uses a frame invariant, geometric approach throughout, which (imo) makes for much simpler and more intuitive representations. It allows you to separate out the internal physics from the effect of choosing a particular coordinate system.
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.
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.
I think it does a very good job of explaining the abstract index notation, which is superficially similar to coordinate notation but conceptually quite different.
I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.
I found the physicist definition of a tensor is actually more confusing, because you are faced with these definitions how to transform these objects, but you never are really explained where does it all come from. While the mathematical definition through differential forms, co-vectors, while being longer actually explains these objects better.
I don't get why people act like this definition is so circular. If you were to explain in detail what "transforms as a second rank tensor" means then it wouldn't be circular anymore. This just isn't the full definition.
> You might find it circular reasoning but it is not
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".
Do you really need me to explain the distinction between "circular" and "recursive", and the nature of HN comments as an informal medium with an audience that generally does understand these things and does not need to have them explicitly called out?
To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way <....explanation of the specific characteristics that mean that a tensor transforms in a way that's independent of the chosen coordinate system...>".
When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.
My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that 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, which is pretty nifty.
> 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.
> 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.
That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order tensors. Secondly the representation is the thing that changes, but the meaning of that representation in the old basis and the new basis is the same. For example, if I take the displacement from me to the top of the Eiffel tower, I can represent that in xyz Cartesian coordinates or in spherical or cylindrical coordinates, or I can measure it relative to an origin that starts with me or at sea level at 0 latlong. The representation will be very different in each case, but the actual displacement from me to the top of the Eiffel tower doesn't change. What has happened is the basis vectors transform in exactly such a way as to make that happen. It's a rank 1 tensor in 3 dimensions because there is a magnitude and one direction (one set of 3 basis vectors) in whatever case.
Now if I want an example of a rank 2 tensor think about a stress tensor. I have a steel beam which is clamped at both ends and a weight is on top of it. This is a tensor field. For every point in the beam there are different forces acting in each direction. So you could imagine the beam as made up of a grid of little rubik's cubes. On each face of each cube you have different net forces. (eg at the middle of the beam the forces are mainly downwards due to gravity, at the ends of the beam the fact that the middle of the beam is bowing downards will lead to the "faces" that point to the middle of the beam to be being pulled towards the middle (transverse to the beam and slightly downwards) whereas the opposite face is pulled in the opposite direction because the ends of the beam are clamped. So I need two sets of basis vectors. One set indicates the "face" experiencing the force, one set indicates the direction of the force. Now just like the vector/rank one tensor case I can represent those in whatever coordinate system I want, and my representation will be different in each case, but will mean the same sets of forces in the same directions and applied to the same directions because both sets of basis vectors will transform to make that true. I would call that a rank 2 tensor field because I would express it as a function from a set of spatial coordinates to a thing which has a magnitude and 2 directions (that's what I think of as the tensor). However I understand physicists and civil engineers and stuff just call the whole thing the stress tensor (not the stress tensor field). I could be wrong.
So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe (eg the displacement from me to the tower or the stress in the beam). From a mathematical point of view I agree of course, mathematical objects themselves are purely arbitrary and abstract. But if you have a bridge and you want to make sure it doesn't buckle and fall down, the stress tensor in the bridge is a real and important fact of the universe that you need to have a decent understanding of.
Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.)
> So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe
OK, but then "the reality of a tensor" not depending on the coordinate system has nothing to do with tensors, and becomes a vacuous observation. It is simply a fact that actual physical quantities don't depend on how you write them down, and hence don't change when you write them down in different ways.
No it’s very important for physics to have a mathematical object that doesn’t change so that you can represent these characteristics of the universe that don’t change. For every observer in every reference frame even though they will use different basis vectors and different components, the combination of basis vectors and components will be the same. That’s extremely powerful. Try the video I linked a few posts above for what I think is a really excellent explanation of what a tensor is (using practical household objects to illustrate everything practically). I think you’ll get it.
I don't understand them. None of the introductory materials I've read on them ever made any sense to me. I kinda sorta understand "state" and "maybe" but the general idea that links these two things together and (I am given to understand) leads to other cool stuff just eludes me.
Easiest way to understand is to write code using them IMO. (Writing a parser for some language with a parser-combinator library is a good place to start.) Otherwise it's like trying to understand how to play a board game by reading the rulebook. It's easier to just try playing the game
Right, but if you fill in the shorthand there’s no reason to think it’s circular; it’s just a normal definition at that point, albeit one without much motivation.
But it's not possible to fill in the shorthand unless you already know what it stands for. Hence: the shorthand is not useful for communicating information, only for social signaling.
A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.
In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.
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.
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.
Does the way "manifold" is used when describing subsets of the representational space of neural networks (e.g. "data lies on a low-dimensional manifold within the high-dimensional representation space") actually correspond to this formal definition, or is it just co-opting the name to mean something simpler (just an embedded sub-space)?
If it is the formal definition being used, then why? Do people actually reason about data manifolds using "atlases" and "charts" of locally euclidean parts of the manifold?
It's hard to prove rigorously which is why people usually refer to it as the "manifold hypothesis." But it is reasonable to suppose that (most) data does live on a manifold in the strict sense of the term. If you imagine the pixels associated with a handwritten "6", you can smoothly deform the 6 into a variety of appearances where all the intermediate stages are recognizable as a 6.
However the embedding space of a typical neural network that is representing the data is not a manifold. If you use ReLU activations the kinks that the ReLU function creates break the smoothness. (Though if you exclusively used a smooth activation function like the swish function you could maintain a manifold structure.)
People also apply the notion of data manifold to language data (which is fundamentally discrete), and even for images the smoothness is hard to come buy (e.g., "images of cars" is not smooth because of shape and colour discontinuities). I guess the best we can do is to hope that there is an underlying virtual "data manifold" from which our datapoints have been "sampled", and knowing its structure may be useful.
Those are less problematic than you might imagine.
- For language, individual words might be discrete, but concepts being communicated have more nuance and fill in the gaps.
- For language, even to the extent that discreteness applies, you can treat the data as being sampled from a coarser manifold and still extract a lot of meaningful structure.
- Images of cars are more continuous than you might imagine because of hue differences induced by time of day, camera lens, shadows, etc.
- Images of cars are potentially smooth even when considering shape and color discontinuities. Manifolds don't have to be globally connected. Local differentiability is usually the thing people are looking for in practical applications.
There's a field known as information geometry. I don't know much about it myself as I'm more into physics, but here's a recent example of applying geometrical analysis to neural networks. Looks interesting as they find a phenomenon analogous to phase transitions during training
Information Geometry of Evolution of Neural Network Parameters While Training
The closest thing that you may get is a manifold + noise. Maybe some people thing about it in that way. Think for example of the graph of y=sin(x)+noise, you can say that this is a 1 dimensional data manifold. And you can say that locally a data manifold is something that looks like a graph or embedding (with more dimensions) plus noise.
But i am skeptical whether this definition can be useful in the real world of algorithms. For example you can define things like topological data analysis, but the applications are limited, mainly due to the curse of dimensionality.
Sometimes statistical rates for empirical risk minimization can be related to the intrinsic dimension of the data manifold (and noise level if present). In such cases, you are running the same algorithm but getting a performance guarantee that depends on the structure of the data, stronger when it is low dimensional.
I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
Manifold: Any m dimensional hyperplane embedded in an n dimensional Euclidean space, where m is less than or equal to n. More simply put, a manifold is any set that can be continuously parameterized, with the number of parameters being the dimension of the manifold.
A continuous manifold will have a line element that allows you to compute distances between its points using its parameters. The simplest line element was first written down by Pythagorus I think, it allows you to compute the distance between two points in a flat manifold. In physics we do away with gravitational forces by realizing that masses move along geodesics (shortest paths) of a manifold, hence the saying,"matter tells spacetime how to curve and spacetime tells matter how to move". We stich together large curvy manifolds like a patch quilt from the locally Euclidean tangent spaces that we erect at any point.
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.
Thanks. I've thought about those possibilites, but I really don't know the reasons.
> 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.
Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with distance from the poles (i.e., the distance is much greater at the equator than the poles), but there is a unique coordinate for every point on the sphere.
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.
> this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0).
I'm guessing that the issue is that I don't know your definition of 'continuous'.
I believe every point on the planet (sphere, for simplification) has unique corresponding coordinates on the map projection (chart). The only exceptions I can see are, A) surfaces perpendicular to the aspect (perspective) of the projection, which is usually straight down and causes points on exactly vertical surfaces to share coordinates; B) if somehow coordinates are limited in precision or to rational numbers; C) some unusual projection that does it.
> 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.
There are cartographic projections that use two charts. Regarding those with one, where is the discontinuity in a Mercator projection? I think when I understand your meaning, it will be clear ...
Continuity is fundamentally a topological property of a mapping. It just means that for a mapping F and a point p, for any neighborhood del of F(p), we can find a neighborhood eps of p such that F(eps) is contained entirely in del. In simpler terms, if you draw a little ball around F(p), I can find a little ball around p whose image under F is contained in the little ball you drew around F(p). If I have coordinates on the sphere that suddenly jump between 0 and 2*pi, I can’t satisfy this property, because points that are arbitrarily close on the sphere will be mapped to opposite sides of the “coordinate square” with sides [0,2*pi).
The Mercator projection is obtained by removing two points from the sphere (both poles) and stretching the hole at each pole until the punctured sphere forms a cylinder, then cutting the cylinder along a line of longitude. So you can see that the 3 discontinuities in the Mercator projection correspond to the top and bottom edges (where we poked a hole at each pole) and the left/right edges (where we cut the cylinder). (Note that stretching the sphere at the poles changes the curvature, but cutting the cylinder does not. The projection would have the same properties on a cylinder.)
It is possible to continuously map the sphere to the entire (infinite) plane if you just remove a single point (the north pole): place the sphere so the south pole is touching the origin of the plane and for any point on the sphere, draw a line from the north pole through that point. Where that line intersects the plane is that point’s image under this mapping (called the Riemann sphere).
That makes sense. As I thought, I just needed to understand continuity in this context. That also helps address my original question - why manifolds aren't widely used in cartography. Thank you.
Every time I try to get some handle on the essence of this topic I fail. No different here. In the second paragraph it defines manifolds as "... shapes that look flat to an ant living on them, even though they might have a more complicated global structure"
So manifolds are complicated shapes that are at large enough a scale that an ant (which species?) will think they're flat....ok
Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.
Its unfortunately all too common for Physics/Math to be taught in that way (extremely technical, memorizing or knowing equations and derivations). The best teachers would always give a ton of context as to why and how these came about.
I just looked it up because I was interested in their etymologies, but it seems that the words actually have the same (Old English/Germanic) root: essentially a portmanteau of "many" + "fold."
This has always caused me trouble when learning new concepts. A name for something will be given (e.g. manifold) and it sounds very much like something that I've come across before (e.g. a manifold in an engine) - and that then gets cemented in my brain as a relationship which I find extremely difficult to shake - and it makes understanding the new concept very challenging. More often than not the etymology of the term is not provided with the concept - not entirely unreasonable, but also not helpful for me personally.
It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!
I thought those are ethymologically about the thin-walled containment of a volumetric interior space where said space is connected to only specific ports/holes, and is often but not necessarily mandatory intertwined with a second such containment for a second space (intake+exhaust).
What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?
The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.
Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings.
This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon
I'm pretty confident I'm not going to give a definition that's suitable for you and I don't wish to engage with diminutive minutiae or arcane jargon in a hostile combative atmosphere.
> Can anyone who is not a mathematician tell me one thing they learned from this?
I can share my two take-aways.
- in the geometric sense, manifolds are spaces analogous to curved 2d surfaces in 3d that extend to an arbitrary number of dimensions
- manifolds are locally Euclidean
If I were to extrapolate from the above, i'd say that:
- we can map a Euclidean space to every point on a manifold and figure out the general transformation rules that can take us from one point's Euclidean space to another point's.
- manifolds enable us to discuss curved spaces without looking at their higher-dimension parent spaces (e.g. in the case of a sphere surface we can be content with just two dimensions without working in 3d).
Naturally, I may be totally wrong about all this since I have no knowledge on the subject...
> 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 learners will do it without learning anything about the writing system.
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.
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/2 way around the equator, turn 90 degrees again. Walk back to the pole. Now the triangle sums 360 degrees!
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.
What's the relation between the different Lee manifolds? Is it a sequence you're supposed to read in order?
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.
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.
Tbh, I never quite understood the appeal of John M. Lee's book. It's not bad but I didn't find it great, either, especially (IIRC) in terms of rigor. Meanwhile, the much less well-known "Manifolds and Differential Geometry" by Jeffrey M. Lee (yeah, almost the same name) was much better.
This is a very informative article about the history of manifolds and their significance. Don’t let the title fool you into this being just a definition.
It’s actually much more well written than the majority or articles we usually come across.
And they have a RSS feed, although it's a bit tricky to figure out, since the relevant header tag for that is set up incorrectly, pointing to a useless empty "comments" feed even from their main page. The actual feed for articles is https://www.quantamagazine.org/feed/
Nice find, thank you. Your sleuthing is appreciated.
Oh dope. Added to my feedbin!
Is that really a good article? I thought it was average. It had some big flaws but was probably still informative for readers with no mathematical knowledge in the domain.
For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as a square [O,2π[²?
I also expected more talk about atlases. In simple cases, it's easy to think of a surface as a deformation of a flat shape, so a natural idea is to think of having a map from the plan to the surface. But, even for a simple sphere, most surfaces can't map to a single flat part of the plan, and you need several maps. But how do you handle the parts where the maps overlap? What Riemmann did was defining properties on this relationship between manifold points and maps (which can be countless).
BTW, I know just enough about relativity to deny that "space-time [is] a four-dimensional manifold", at least a Riemmannian manifold. IIRC, the usual term is Minkowski-spacetime.
> Precisely, why more as a manifold than as a square
In a double pendulum, each arm can freely rotate (there is no stopping point). This means 0 degrees and 360 degrees are the same point, so the edges of the square are actually joined. If you join the left and right edges to each other, then join the top and bottom edges to each other, you end up with a torus.
> Precisely, why more as a manifold than as a square [O,2π[²?
Because, as the article explains, it's a torus (loop crossed with a loop), not a square (segment crossed with a segment).
Minkowski spacetime is the term in special relativity, i.e. the flat case, or zero curvature. In general relativity, spacetime is a pseudo Riemannian manifold, like the sibling comment says. Unlike Minkowski spacetime, it can be curved.
> BTW, I know just enough about relativity
Unfortunately this is one of those things where that knowledge is not enough.
The GR model of spacetime is that it is locally Minkowski but globally a manifold of Minkowski patches.
Spacetime is a four-dimensional manifold (at least theoretically - who knows what it is in reality). Technically it's a pseudo-Riemannian manifold since the metric is not positive definite: it can be negative or zero for non-zero vectors. A Riemannian manifold proper has a positive definite metric, but in popularizations like this I wouldn't really expect them to get into these kinds of distinctions.
I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles.
Besides not treating readers like idiots, they take themselves seriously, hire smart people, tell good stories but aren't afraid to stay technical, and simply skip all the clickbait garbage. Right now from the Scientific American front page: "Type 1 Diabetes science is having a moment". Or from Nature: "'Biotech Barbie' says ..". Granted I cherry-picked these offensive headlines pandering to facebook/twitter from many other options that might be legitimately interesting reads, but on Quanta there's also no paywalls, no cookie pop-ups, no thinly-veiled political rage-baiting either
Quanta is amazing because it doesn't have to worry about money. It's a publication run by the Simons Foundation, funded with the proceeds of the wildly successful RenTec hedge fund. So they get pretty much full editorial control.
For other publications they are beholden to people who haven't figured out ad-block, and your bar needs to be pretty low to capture that revenue.
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.
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?
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.
IMO the most common reason why something is presented in a more complex way is that it is badly explained.
Of course, most common or not, each case is different.
Also speaks to a lack of understanding on the author's part; people who truly understand some subject are generally much more adept at explaining it in simpler terms – ie without adding complexity beyond the subject's essential complexity
I don’t see how that is beneficial. If a simple concept relates to a complex one then explain the complexity, don’t add it.
> If a simple concept relates to a complex one then explain the complexity, don’t add it.
So write a text of at least 500 pages to explain the complexity. :-)
It's OK to keep going deeper into the material if you aren't tired yet.
It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.
Good game and a hard question, especially if you make "comparable" more explicit. I'd add "noncommercial, open-access", and "modern" in the sense that it happened under the current norms with respect to legacy and the social contract.
Carnegie Libraries, Nobel Prizes, Rhodes scholarships?
Clay Mathematics Institute, with its 7 Millennium problems
Good one!
Stanford & Carnegie Mellon universities...
I like what MacKenzie Scott is doing in the space of "What it takes to actually spend down billions of dollars"
Clay Mathematics Institute, with its 7 Millennium problems
Mathematica?
I agree. I find their articles very enjoyable. And even though they stay technical, they don’t descend into becoming a technical journal. The content is still accessible to a non-expert like me.
Agreed. I'm not a mathematician - and to me a manifold is more familar in the context of engines. But I found both the text and the diagrams very useful.
When you use the word "engine" on HN, it can be understood as many things that aren't what you think (e.g. game engines).
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 visualize the geometry which is actually hard and sometimes counter intuitive.
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 huge amount of attention is typically given to the Lorenz transformations required when coordinates change. However, the (Minkowski) space that is the setting for special relativity is well defined without reference to any particular coordinate system, as an affine space with a particular (pseudo-)metric. It's not conceptually very complicated, and I never properly understood special relativity until I saw it explained in those terms in the amazing book Special Relativity in General Frames by Eric Gourgoulhon.
For tensors, the basis-independent notion is a multilinear map from a selection of vectors in a vector space and forms (covectors) in its dual space to a real number. The transformation properties drop out of that, and I find it much more comfortable mentally to have that basis-independent idea there, rather than just coordinate representations and transformations between them.
I agree that focusing on Lorentz transformations is the wrong way to approach thinking about special relativity. But It might be the right way to teach it to physics students.
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.
The basics of special relativity came up in my first year of university, and the rest didn't really get focused on until my second year.
The first time around I was still encountering linear algebra and vector spaces, while for the second I was a lot more comfortable deriving things myself just given something like the Minkowski "inner product".
(As an aside: I really love abstract index notation for dealing with tensors)
> 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.
> But It might be the right way to teach it to physics students.
Having studied physics, I would disagree rather strongly. I only really started understanding Special Relativity once I had a clear understanding of the math. (And then it becomes almost trivial.) Those of my fellow class mates, however, who didn't take the time to take those additional (completely optional) math classes, ended up not really understanding it at all. They still got confused by what it all meant, by the different paradoxes, etc.
I saw the same effect when, later, I was a teaching assistant for a General Relativity class.
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.
Mine was similar actually, just in Ireland.
I had started in a theoretical physics degree which was jointly taught by the maths and physics department. By my final year I had changed into an ostensibly pure maths degree, although I did it mainly to take more advanced theoretical/mathematical physics courses (which were taught by the maths department), and avoid having to do any lab work—a torsion pendulum experiment was my final straw on that one, I don't know what caused it to fuck up, but fuck that.
In the end I took on more TP courses than the TP students, nearly burnt out by the end of the year, and... didn't exactly come out with the best exam results.
Taylor & Wheeler's Spacetime Physics is similar. They emphasize the importance of frame invariant representations. (I highly recommend the first edition over the second edition, the second edition was a massive downgrade.)
Kip Thorne was also heavily influenced by this geometric approach. Modern Classical Physics by Thorne & Blandford uses a frame invariant, geometric approach throughout, which (imo) makes for much simpler and more intuitive representations. It allows you to separate out the internal physics from the effect of choosing a particular coordinate system.
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.
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.
Weinberg ≠ Wald. Wald's book is great! (For GR, of course, not SR.)
Indeed! I meant that it's good to know Wald is mathematically modern and not encrusted with coordinates. Saves me buying another book :-D
(The comment I replied to mentioned both.)
I think it does a very good job of explaining the abstract index notation, which is superficially similar to coordinate notation but conceptually quite different.
I think _Spacetime Physics_ takes roughly the same approach (they call it “the invariant interval”), but with much less mathematical sophistication required.
Thanks for the book recommendation.
I found the physicist definition of a tensor is actually more confusing, because you are faced with these definitions how to transform these objects, but you never are really explained where does it all come from. While the mathematical definition through differential forms, co-vectors, while being longer actually explains these objects better.
I don't get why people act like this definition is so circular. If you were to explain in detail what "transforms as a second rank tensor" means then it wouldn't be circular anymore. This just isn't the full definition.
> You might find it circular reasoning but it is not
Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means.
Of course, there are some definitions that reference themselves that are not circular, or alternatively, some circular definitions that are useful. For example, the definition of "ancestors" might be "your parents and your parents' ancestors" (with the implication that if you don't exist, you don't have any ancestors). But I agree that this is not a useful definition for "tensor".
> "your parents and your parents' ancestors"
That's recursive, not circular. These are not the same thing despite being closely related. Recursive definitions are useful. Circular ones are not.
I was using the definition you gave for “circular”
Do you really need me to explain the distinction between "circular" and "recursive", and the nature of HN comments as an informal medium with an audience that generally does understand these things and does not need to have them explicitly called out?
To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way <....explanation of the specific characteristics that mean that a tensor transforms in a way that's independent of the chosen coordinate system...>".
When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.
My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that 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, which is pretty nifty.
https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...
> 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.
> 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.
That's just incorrect though for a couple of reasons. Firstly, a vector in the sense in which it is used in physics is a rank 1 tensor so it has this transformation behaviour just like other higher order tensors. Secondly the representation is the thing that changes, but the meaning of that representation in the old basis and the new basis is the same. For example, if I take the displacement from me to the top of the Eiffel tower, I can represent that in xyz Cartesian coordinates or in spherical or cylindrical coordinates, or I can measure it relative to an origin that starts with me or at sea level at 0 latlong. The representation will be very different in each case, but the actual displacement from me to the top of the Eiffel tower doesn't change. What has happened is the basis vectors transform in exactly such a way as to make that happen. It's a rank 1 tensor in 3 dimensions because there is a magnitude and one direction (one set of 3 basis vectors) in whatever case.
Now if I want an example of a rank 2 tensor think about a stress tensor. I have a steel beam which is clamped at both ends and a weight is on top of it. This is a tensor field. For every point in the beam there are different forces acting in each direction. So you could imagine the beam as made up of a grid of little rubik's cubes. On each face of each cube you have different net forces. (eg at the middle of the beam the forces are mainly downwards due to gravity, at the ends of the beam the fact that the middle of the beam is bowing downards will lead to the "faces" that point to the middle of the beam to be being pulled towards the middle (transverse to the beam and slightly downwards) whereas the opposite face is pulled in the opposite direction because the ends of the beam are clamped. So I need two sets of basis vectors. One set indicates the "face" experiencing the force, one set indicates the direction of the force. Now just like the vector/rank one tensor case I can represent those in whatever coordinate system I want, and my representation will be different in each case, but will mean the same sets of forces in the same directions and applied to the same directions because both sets of basis vectors will transform to make that true. I would call that a rank 2 tensor field because I would express it as a function from a set of spatial coordinates to a thing which has a magnitude and 2 directions (that's what I think of as the tensor). However I understand physicists and civil engineers and stuff just call the whole thing the stress tensor (not the stress tensor field). I could be wrong.
So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe (eg the displacement from me to the tower or the stress in the beam). From a mathematical point of view I agree of course, mathematical objects themselves are purely arbitrary and abstract. But if you have a bridge and you want to make sure it doesn't buckle and fall down, the stress tensor in the bridge is a real and important fact of the universe that you need to have a decent understanding of.
> That's just incorrect though
Quite possible. But that's in no small measure because I have yet to find an actual cogent definition of "tensor" that distinguishes a tensor from an array. (I have a similar problem with monads.)
> So what I mean when I talk about the reality of the tensor I mean whatever it is the tensor is expressing in the physical universe
OK, but then "the reality of a tensor" not depending on the coordinate system has nothing to do with tensors, and becomes a vacuous observation. It is simply a fact that actual physical quantities don't depend on how you write them down, and hence don't change when you write them down in different ways.
No it’s very important for physics to have a mathematical object that doesn’t change so that you can represent these characteristics of the universe that don’t change. For every observer in every reference frame even though they will use different basis vectors and different components, the combination of basis vectors and components will be the same. That’s extremely powerful. Try the video I linked a few posts above for what I think is a really excellent explanation of what a tensor is (using practical household objects to illustrate everything practically). I think you’ll get it.
Could you elaborate on your problem with monads?
I don't understand them. None of the introductory materials I've read on them ever made any sense to me. I kinda sorta understand "state" and "maybe" but the general idea that links these two things together and (I am given to understand) leads to other cool stuff just eludes me.
Easiest way to understand is to write code using them IMO. (Writing a parser for some language with a parser-combinator library is a good place to start.) Otherwise it's like trying to understand how to play a board game by reading the rulebook. It's easier to just try playing the game
Right, but if you fill in the shorthand there’s no reason to think it’s circular; it’s just a normal definition at that point, albeit one without much motivation.
But it's not possible to fill in the shorthand unless you already know what it stands for. Hence: the shorthand is not useful for communicating information, only for social signaling.
A manifold is a surface that you can put a cd shaped object on in any place on the surface, you can change the radius of the cd but it has to have some radius above 0.
Nicely done!
Initially I recoiled at the thought of the stiffness of the CD, but of course your absolutely right, at least for 2d manifolds.
Including the hole at the center of the CD?
> you can put a cd shaped object on
You're thinking of open sets.
In particular, consider two intersecting planes. You can put all the discs you like on that surface, but it's not a manifold because on the line of intersection it's not locally R2.
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.
>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?
> 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.
Thanks a lot!
> 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.
Does the way "manifold" is used when describing subsets of the representational space of neural networks (e.g. "data lies on a low-dimensional manifold within the high-dimensional representation space") actually correspond to this formal definition, or is it just co-opting the name to mean something simpler (just an embedded sub-space)?
If it is the formal definition being used, then why? Do people actually reason about data manifolds using "atlases" and "charts" of locally euclidean parts of the manifold?
It's hard to prove rigorously which is why people usually refer to it as the "manifold hypothesis." But it is reasonable to suppose that (most) data does live on a manifold in the strict sense of the term. If you imagine the pixels associated with a handwritten "6", you can smoothly deform the 6 into a variety of appearances where all the intermediate stages are recognizable as a 6.
However the embedding space of a typical neural network that is representing the data is not a manifold. If you use ReLU activations the kinks that the ReLU function creates break the smoothness. (Though if you exclusively used a smooth activation function like the swish function you could maintain a manifold structure.)
People also apply the notion of data manifold to language data (which is fundamentally discrete), and even for images the smoothness is hard to come buy (e.g., "images of cars" is not smooth because of shape and colour discontinuities). I guess the best we can do is to hope that there is an underlying virtual "data manifold" from which our datapoints have been "sampled", and knowing its structure may be useful.
Those are less problematic than you might imagine.
- For language, individual words might be discrete, but concepts being communicated have more nuance and fill in the gaps.
- For language, even to the extent that discreteness applies, you can treat the data as being sampled from a coarser manifold and still extract a lot of meaningful structure.
- Images of cars are more continuous than you might imagine because of hue differences induced by time of day, camera lens, shadows, etc.
- Images of cars are potentially smooth even when considering shape and color discontinuities. Manifolds don't have to be globally connected. Local differentiability is usually the thing people are looking for in practical applications.
There's a field known as information geometry. I don't know much about it myself as I'm more into physics, but here's a recent example of applying geometrical analysis to neural networks. Looks interesting as they find a phenomenon analogous to phase transitions during training
Information Geometry of Evolution of Neural Network Parameters While Training
https://arxiv.org/abs/2406.05295
The closest thing that you may get is a manifold + noise. Maybe some people thing about it in that way. Think for example of the graph of y=sin(x)+noise, you can say that this is a 1 dimensional data manifold. And you can say that locally a data manifold is something that looks like a graph or embedding (with more dimensions) plus noise.
But i am skeptical whether this definition can be useful in the real world of algorithms. For example you can define things like topological data analysis, but the applications are limited, mainly due to the curse of dimensionality.
Sometimes statistical rates for empirical risk minimization can be related to the intrinsic dimension of the data manifold (and noise level if present). In such cases, you are running the same algorithm but getting a performance guarantee that depends on the structure of the data, stronger when it is low dimensional.
Lobachevsky... "the analytic and algebraic topology of locally Euclidean metrizations of infinitely differentiable Riemannian manifolds"
There's antimony, arsenic, aluminum, selenium…
Plagiarize!
боже мой.
> боже мой.
Pokemon?
You must not know great Lobachevsky.
I always found interesting that the English mathematical terminology has two different names for "stuff that locally looks like R^n" (manifold) and "stuff that is the zero locus of a polynomial" (variety). Other languages use the same word for both, adding maybe an adjective to specify which one is meant if not clear from the context. In Italian for example they're both "varietà"
In English, not all varieties are manifolds, see forex https://math.stackexchange.com/a/9017/120475
FTA
> The term “manifold” comes from Riemann’s Mannigfaltigkeit, which is German for “variety” or “multiplicity.”
This is not really something limited to mathematics.
Manifold: Any m dimensional hyperplane embedded in an n dimensional Euclidean space, where m is less than or equal to n. More simply put, a manifold is any set that can be continuously parameterized, with the number of parameters being the dimension of the manifold.
A continuous manifold will have a line element that allows you to compute distances between its points using its parameters. The simplest line element was first written down by Pythagorus I think, it allows you to compute the distance between two points in a flat manifold. In physics we do away with gravitational forces by realizing that masses move along geodesics (shortest paths) of a manifold, hence the saying,"matter tells spacetime how to curve and spacetime tells matter how to move". We stich together large curvy manifolds like a patch quilt from the locally Euclidean tangent spaces that we erect at any point.
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.
Thanks. I've thought about those possibilites, but I really don't know the reasons.
> 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.
Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with distance from the poles (i.e., the distance is much greater at the equator than the poles), but there is a unique coordinate for every point on the sphere.
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)
This part I don't grasp:
> this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0).
I'm guessing that the issue is that I don't know your definition of 'continuous'.
I believe every point on the planet (sphere, for simplification) has unique corresponding coordinates on the map projection (chart). The only exceptions I can see are, A) surfaces perpendicular to the aspect (perspective) of the projection, which is usually straight down and causes points on exactly vertical surfaces to share coordinates; B) if somehow coordinates are limited in precision or to rational numbers; C) some unusual projection that does it.
> 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.
There are cartographic projections that use two charts. Regarding those with one, where is the discontinuity in a Mercator projection? I think when I understand your meaning, it will be clear ...
Continuity is fundamentally a topological property of a mapping. It just means that for a mapping F and a point p, for any neighborhood del of F(p), we can find a neighborhood eps of p such that F(eps) is contained entirely in del. In simpler terms, if you draw a little ball around F(p), I can find a little ball around p whose image under F is contained in the little ball you drew around F(p). If I have coordinates on the sphere that suddenly jump between 0 and 2*pi, I can’t satisfy this property, because points that are arbitrarily close on the sphere will be mapped to opposite sides of the “coordinate square” with sides [0,2*pi).
The Mercator projection is obtained by removing two points from the sphere (both poles) and stretching the hole at each pole until the punctured sphere forms a cylinder, then cutting the cylinder along a line of longitude. So you can see that the 3 discontinuities in the Mercator projection correspond to the top and bottom edges (where we poked a hole at each pole) and the left/right edges (where we cut the cylinder). (Note that stretching the sphere at the poles changes the curvature, but cutting the cylinder does not. The projection would have the same properties on a cylinder.)
It is possible to continuously map the sphere to the entire (infinite) plane if you just remove a single point (the north pole): place the sphere so the south pole is touching the origin of the plane and for any point on the sphere, draw a line from the north pole through that point. Where that line intersects the plane is that point’s image under this mapping (called the Riemann sphere).
That makes sense. As I thought, I just needed to understand continuity in this context. That also helps address my original question - why manifolds aren't widely used in cartography. Thank you.
Every time I try to get some handle on the essence of this topic I fail. No different here. In the second paragraph it defines manifolds as "... shapes that look flat to an ant living on them, even though they might have a more complicated global structure"
So manifolds are complicated shapes that are at large enough a scale that an ant (which species?) will think they're flat....ok
Man, I wish that the modern internet -- and great stuff like this -- had been around when I took GR way back when. My math chops were never good enough to /really/ get it and there were so many concepts (like this one) that were just symbols to me.
Its unfortunately all too common for Physics/Math to be taught in that way (extremely technical, memorizing or knowing equations and derivations). The best teachers would always give a ton of context as to why and how these came about.
Wikipedia has a thorough intro article https://en.wikipedia.org/wiki/Manifold
Funny how a car manifold is also a mathematical manifold but the word seems to come from different roots.
I just looked it up because I was interested in their etymologies, but it seems that the words actually have the same (Old English/Germanic) root: essentially a portmanteau of "many" + "fold."
This has always caused me trouble when learning new concepts. A name for something will be given (e.g. manifold) and it sounds very much like something that I've come across before (e.g. a manifold in an engine) - and that then gets cemented in my brain as a relationship which I find extremely difficult to shake - and it makes understanding the new concept very challenging. More often than not the etymology of the term is not provided with the concept - not entirely unreasonable, but also not helpful for me personally.
It becomes a bigger problem when the etymology is actually a chain of almost arbitrary naming decisions - how far back do I go?!
On many occasions in my mathematics education I was able to figure out and use a concept based solely on its name. (e.g. Feynman path integral)
Names are important.
I thought those are ethymologically about the thin-walled containment of a volumetric interior space where said space is connected to only specific ports/holes, and is often but not necessarily mandatory intertwined with a second such containment for a second space (intake+exhaust).
So what is the "Not a manifold." part? The actually interesting part.
A very tight poker player
What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?
The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.
Well as a non mathematician all I saw in your description was opaque jargon. "locally euclidean hausdorff topological space" means nothing to me. It'd be like if I asked what the Spanish word "¡hola!" meant and the answer was in evocative Spanish poetry. Extremely unlikely to be helpful to that person who doesn't know basic greetings.
This article breaks that loop and it's refreshing to see a large topic not explained as an amalgamation of arcane jargon
If this article broke some loop, you could answer my question and tell me one thing you got out of it. What's a manifold?
I'm pretty confident I'm not going to give a definition that's suitable for you and I don't wish to engage with diminutive minutiae or arcane jargon in a hostile combative atmosphere.
I don't think I can improve this silence.
Have a good day.
> Can anyone who is not a mathematician tell me one thing they learned from this?
I can share my two take-aways.
- in the geometric sense, manifolds are spaces analogous to curved 2d surfaces in 3d that extend to an arbitrary number of dimensions
- manifolds are locally Euclidean
If I were to extrapolate from the above, i'd say that:
- we can map a Euclidean space to every point on a manifold and figure out the general transformation rules that can take us from one point's Euclidean space to another point's.
- manifolds enable us to discuss curved spaces without looking at their higher-dimension parent spaces (e.g. in the case of a sphere surface we can be content with just two dimensions without working in 3d).
Naturally, I may be totally wrong about all this since I have no knowledge on the subject...
ok but she was talking about riemann
> 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 learners will do it without learning anything about the writing system.
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.
A $1,500 trip to the mechanic
That's why I clicked the title...thought for sure I was getting some engine knowledge
This is such a well written article and the author is such a good communicator. Looks like they've written a book as well called Mapmatics:
[0] https://www.paulinarowinska.com/about-me
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Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/4 the way around the equator, turn 90 degrees again. Then walk back to the pole. A triangle with sum 270 degrees!
See you and raise you.
Stand at one of the poles. Walk to the equator, turn 90 degrees. Walk 1/2 way around the equator, turn 90 degrees again. Walk back to the pole. Now the triangle sums 360 degrees!
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