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.
I have a tip for following lectures (or any technical talk, really) that I've been meaning to write about for a while.
As you follow along with the speaker, try to predict what they will say next. These can be either local or global predictions. Guess what they will write next, or what will be on the next slide. With some practice (and exposure to the subject area) you can usually get it right. Also try to keep track of how things fit into the big picture. For example in a math class, there may be a big theorem that they're working towards using lots of smaller lemmas. How will it all come together?
When you get it right, it will feel like you are figuring out the material on your own, rather than having it explained to you. This is the most important part.
If you can manage to stay one step ahead of the lecturer, it will keep you way more engaged than trying to write everything down. Writing puts you one step behind what the speaker is saying. Because of this, I usually don't take any notes at all. It obviously works better when lecture notes are made available, but you can always look at the textbook.
People often assume that I have read the material or otherwise prepared for lectures, seminars, etc., because of how closely I follow what the speaker is saying. But really most talks are quite logical, and if you stay engaged it's easy to follow along. The key is to not zone out or break your concentration, and I find this method helps me immensely.
This is fun to do during lectures but in my experience only about 5-10% of my learning happened in math class. The other 90% happened at home as I worked through the problem sets.
Essentially the lectures served as an inefficient way of delivering me a set of notes which I’d then reference during homework sessions. I could often predict what was coming next in the lecture but the really hard parts were the key parts in some technical lemmas that were necessary to complete the theorem. Learning how to figure out a key step like that had to come completely on my own (with no spoilers).
In a lot of ways, math lectures really started to turn into an experience similar to watching a Let’s Play of a favourite video game. Watching those can tell you exactly what you need to do to get past the part where you’re stuck but they don’t in general make you better at video games. For that you need to actually play them yourself.
I had a math professor in college that would often say to our class, "You cannot be like Michael Jordan by just watching Michael Jordan. If you want to be better at basketball, you have to practice. Math is no different." No matter how you spin it, he was correct -- unless you are like Ramanujan and a Hindu god just reveals a solution to you.
Honestly though, I believe I learn better in a similar manner to what you described. I would rather just read the textbook and learn on my own. I find that to be a far more efficient learning style for me. However, I typically always went to class for a handful of reasons:
1. To signal that I cared about the subject to the professor (whether I honestly cared or not). Though I had some classes that actually penalized a lack of attendance.
2. There is comradery in group struggle. It was nice way to meet other students that had a common goal. I made many friends during my time. Some of which I still keep in touch with a decade later. In fact, I met my SO in one of my classes -- all because we studied together.
3. The main reason being, I paid for the class, and I wanted to get my money's worth out of it. While passing the course and learning the material was the goal. I'd hate knowing I just paid to teach myself everything. I could have done that for free, so I wanted something more out of the deal.
One of thing I should add is that I am poorly disciplined and have poor executive functioning, so I probably picked up more in class that I would admit -- I didn't have a control to compare against. Still to this date, I rely heavily on solutions to the problems. Not in a way that allows me to cheat, but I would likely be unable to be certain I was teaching myself correctly if I didn't have the answers or know of a method to verify my work. I am confident that I cannot be confident in my answers to nearly anything. I am prone to too many mistakes.
If one goes far enough in math, one will encounter solutions where there are not clear answers and one must use all of their knowledge and abilities to support their answers. And that my YN friends, is why I am not a mathematician despite my love for the subject.
- I find that writing notes in class helped me learn just through the physical action of my hands. (I think there is some formal study of this as a phenomenon). I am poorly disciplined so at least getting that hour or so of writing notes is probably more than I would have managed alone.
- In class, sometimes the lecturer provides helpful intuition for something through informal speech or even intonation. For example I struggled with the concept of ergodicity from a textbook until I saw someone explain it to me like I'm 5. I find that often, textbooks are like man pages, in that they are almost afraid to provide informal/intuitive writing for fear of appearing unserious.
p.s. if ChatGPT existed 30 years ago I would have managed to learn so much more instead of spinning wheels on dry writing. ChatGPT is really good at being a "personalized manpage explainer"
The viewpoint of a lecture as an inefficient note delivery system is a pretty common and reductive view. Your "Let's Play" analogy was pretty apt though.
I think their (potential) value seems pretty clear when you look at language courses -- you can't possibly hope to develop fluency in a language by studying it in isolation from books -- forming your own sentences and hearing how other human beings do the same in real time is pretty decisive.
With math classes, YMMV, especially since they are rarely so interactive at the upper division and graduate level, but at the very least seeing an instructor talk about math and work through problems (and if you are lucky to have a particularly disorganized one, get stuck, and get themselves unstuck) can go a long way. But to be fair I regularly skipped math lectures in favor of reading too, heh
I rarely skipped math lectures in university (only when the prof was really bad; but then I watched video lectures taught by a different prof from a previous term).
The lectures in the hardest math classes I took did not feature any “working through problems.” They were 50 minute pedal-to-the-metal proof speedrun sessions that took me 2-3 hours of review and practice work to fully understand. I don’t know how anyone can see a lecture like that and not see it as an inefficient note delivery system.
I did have math classes where profs worked through problems but those were generally the much easier applied math classes. Those were the ones I least needed to attend lectures for because there you’re just following the steps of an algorithm rather than having to think hard about how to synthesize a proof.
For language learning it’s hard to beat full immersion. When we learn our first language (talking to our parents as children) we don’t learn it by theory (memorizing verb conjugations), we learn it by engaging the language centre in our brains. I think language classes are more useful if you want to learn to write and translate in that language, where you need a strong theoretical background. If your main goal for language learning is being able to speak with loved ones or being able to travel and speak fluently with locals, then sitting in a classroom listening to a lecture seems like a very difficult way to do that.
I meant "problems" in a broad sense -- I loved disorganized professors who would pause and stare at their lecture notes in silence for a minute, realize their proof or example contained some flaw, and then have to correct it on the fly.
I found those moments really valuable if course-correcting was non-trivial -- the typical Definition-Theorem-Proof-Example format certainly is essential for organizing one's thinking and communicating new math in a way that's digestible to other mathematicians, but it is not how mathematicians actually think about math or solve novel problems
In the grad analysis sequence this "course correcting" mechanic was built into the course, since we were required to regularly solve a challenging problem and then present its proof to the class and withstand intense questioning from both the professor and peers. If you caught an error in someone's proof and could help the presenter arrive at a correct proof, you'd both earn points.
The thrill of surviving an incredulous "Wait a second..." from that particular professor (who later became my research advisor) was hard to beat
Anyway my intent was to analogize math lectures (whatever they might look like) with language courses or immersion in the sense that they are an opportunity to practice speaking and listening, and to immerse yourself in cultural norms. I think it goes a bit deeper than this, in that language is inextricably connected to most thought and vice versa -- we experience this in a very explicit way whenever we find our thinking clarified in the process of formulating a question, but it's always there
That said, pure immersion for language learning is actually easy to beat -- lots of research shows that immersion together with explicit grammar instruction has far better learning outcomes than immersion alone. Immersion alone misses lots of nuance -- and it relies on the speaker being acutely aware of the difference between their output and target forms.
With your verb conjugation example, lots of time can be saved by knowing that there's a thing called the subjunctive and that it is distinct from tense and it shows up in a myriad of places tending to concern hypotheticals
Similarly, I gain a lot from talking to mathematicians and attending conferences. But I also need to spend time alone consulting relevant theory, reading papers, and playing with examples. Both are important, but in math it seems you one get away with less immersion
I consider the value in math lectures to come from the speaker’s explanation of why to expect certain things. Is this an obvious fact in another context, rewritten for this application? Is this a surprise? What reasons besides the rigorous argument are there for believing the theory?
A lot of the theorems I learned in school weren’t particularly amenable to intuitive explanations like that.
For example, take Galois theory. The fact that a polynomial’s solvability by radicals depends on the solvability of its Galois group is surprising and not intuitive at all. The fundamental theorem of Galois theory is a very technical result utilizing purpose-built mathematical structures that were developed specifically to study the solutions of polynomial equations.
Agree with this comment but follow up to this tip:
Only use this as a learning technique. Do not accidentally let this bleed over into personal 1:1 conversations.
I know some people in my life who are "smart" and they will cut people off in the middle of conversation to the effect of "oh yeah I already know what you are going to say, let me go ahead and cut you off so I can respond faster".
On top of being completely obnoxious on the face of it, they are wrong enough times in their predictions to where it completely fucks the conversation.
Incompatible communication styles (turn-taking vs. constructive overlapping) can be frustrating. Turn-takers may consider overlappers to be rude and aggressive and try to stop the overlap, which overlappers may in turn consider rude and aggressive; yet full-duplex is more efficient than half-duplex and even basic ACKs are important for reliable communication.
Unfortunately Zoom is 1/N-duplex, which ruins it for everyone.
Good point! I used to be guilty of this myself, so now I'm pretty sensitive about other people doing it. I am now one of the more senior students in an academic research group, and some of the younger members would benefit from this advice. I think it's a symptom of sophomorism, and hopefully most will grow out of it.
I agree it's especially frustrating when they don't even get it right. That crosses the line for me, and I will admonish them to let me finish what I am saying.
Well said. And it makes sense, if you define intelligence as the ability to successfully predict the future.
And how interesting that that is literally how LLMs are trained during pretraining. Like Ilya said: To predict the name revealed as the murderer at the end of a detective novel, you must have followed the plot, have world knowledge about physics, psychology, etc..
And that’s what you’re pointing at here. Testing yourself on the ability to predict during a lecture is like running a loss function to keep you on your toes.
Oh yea. A good detective novel gives the reader all the necessary information to know the answer. Many lousier novels just keep some essential information hidden until the monologue, because they haven’t got a tricky enough mystery, and really shitty ones just accidentally reveal it, often by over-using tropes or having silly patterns like “it’s always the dark and brooding guy”. Ever read any Dan Brown? In Angels and Demons he gives it away on the first page with an anagram.
That monologue curtain pull is a hallmark of Sherlock Holmes. I wouldn’t call Doyle’s books lousy for that though. Creating brain-teasers for the audience isn’t really the point. It’s all about a character Doyle found insufferable but the audience loved!
If that is true, which sometimes it is, you having suspected about anyone except the actual murderer, and at the same time it is totally obvious who the culprit is on a second reading, then it is a very good plot.
People who read detective novels do. Every other book the clues are not the point and so they are skipped - or more likely there is no investigation, you are just told and the plot moves on.
Which is why I hate the PowerPoint presentation-based lectures. Speaker typically goes too fast, and their brain does not actually break down the arguments into logical steps. They just read the slides.
Reading from slides is the absolute worst way of delivery anywhere, whether it’s a lecture or an internal presentation to your work team, doesn’t matter. The best power point slides have zero overlapping words with what the presenter is saying except perhaps some slide or section titles.
Chalk and board though is not necessarily the best. Power point supports magic hotkeys - B and W - and allows drawing on slides. When done properly with a stylus, it’s incrementally better in almost every way than chalk, though a proper lecture hall with multiple blackboards will still hold its own.
I slightly disagree. Specifically, in the case of academic lectures, on:
> The best power point slides have zero overlapping words with what the presenter is saying except perhaps some slide or section titles.
Especially when not taking notes, having the slides effectively be lecture notes is great to allow you to go back to the content of the lecture days or weeks later.
That does not mean that I want a lecturer to just read from the slides. But I want the slides to be more than just a visual aid for the lecture. They need to be reference material as well. This is also generally accepted, and can be considered the reason so many other presentations where this is a bad idea, still have the bad lecture-style slides. Because its what is modeled to people during their education.
Note, for presentations to stakeholders, or presentations of results, or almost any other type of professional presentation. Slides should probably be visual aids only, and not reference material. But lectures are a special case.
> having the slides effectively be lecture notes is great...
This is usually considered a great sin by presentation gurus, even for lectures. For academic material, there would hopefully be a textbook as a reference material.
Although I would prefer the slides having no overlapping words, but doing it this way is very punishing for whoever that skip the class for whatever reason.
Looking at the slides no longer build enough context for you to learn and catchup on what you have missed, so in the end every lecture is unskippable.
I strongly agree - I started teaching from slides and shifted more and more to blackboard and chalk.
I found that they can also be combined well by showing a slide that gives some guidance where "we" are in the material, supplemented by writing and drawing on the blackboard to explain one or more bullet points or statements from the slides, and to answer student questions.
This also forces students to take additional notes, which helps them per muscle memory.
"... when people converse or share an experience, their brain waves synchronize. Neurons in corresponding locations of the different brains fire at the same time, creating matching patterns ..."
"The experience of “being on the same wavelength” as another person is real, and it is visible in the activity of the brain."
"Synchrony may be a sign of shared cognitive processing ..."
"The mouse study suggested another level of meaning for synchrony: it predicts the outcomes of future interactions."
This is exactly how I read research papers and how I advocate others read them as well. As you read try to figure out how you would solve the problem outlined and what experiments you would need to perform.
That may require some experience, which you can obtain gradually by reading lots of papers, some good, some bad, to see how they do it. Eventually, you can tell if an experiment is suitable for showing strong support of the RQ or not.
(And you will also learn to read between the lines e.g. "Our resulst are PROMISING..." = there is much space for improvement etc.)
It meant that I understood the scaffolding of the course: the broad goals of the subject, the main ways to tackle them, what is currently being explored, and why are we going in this direction at this moment.
There was one class where this started to fail, which was a class without slides or a book. This meant that, without notes, when I recalled a proof technique but not the details, I had to resort to asking others for their notes. Because it wasn't documented anywhere else.
There were times in university where I had figured out the material on my own (maybe even several lectures ahead), and the confirmation actually felt a bit disappointing.
Great advice! Personally, I got immense value from writing notes but never when I wrote them during the lecture. 30 minutes after the lecture has ended is a perfect time time to sit down in the library and write notes for what the lecture was about. This gives enough time to reflect about the big picture, but not so too much time that the details are lost.
I like this idea but I always struggled to keep up with note taking. And the teachers were struggling to get through all the material. There was a race on both sides! But that was many years ago. If I were doing it today, I'd take pictures with my phone, use a computer to transcribe it, and then I'd have enough time to do what you said.
My technique was to write tons of notes during the lecture. In college, I would have many pages of notes for each lecture and because writing is more of an active process than just sitting or spacing out for an hour, I rarely had to study for an exam.
This is good advice for the LSAT too, and baked into LSAT Demon's app. If you can predict an answer before looking at the choices, you're probably on the right track.
Yes but if you don’t know the answer by the time the light goes on (the question is finished read), you will never get in. And if you buzz in without knowing the answer you will lose points. So you have to know the answer before the light goes , then be ready to buzz as soon as eligible. Jeopardy is a good example.
Every learning method you can think of has been thought of before and all variations have been implemented in classrooms throughout time. It is mostly pseudo-science. You either put in the effort to learn and struggle until you succeed or you don't. There is no secret sauce.
I've met lot of smart guys never getting anywhere, because they were always looking for a shortcut and not to do the real work.
Linux instructor Jason Canon wrote once that there's a lot of people who spend 90% of the time reading articles on how to learn Linux, but only 10% really practicing.
OTOH it's a really cool way to stay focused and engaged with the lecture.
I think a lot of writing online about productivity is like this. Some people seem to have a near endless appetite for writing on pens and notebooks, note taking systems, text editors, desk accesssories, every day carry, etc…
I've seen this a lot over the years and I've been guilty of it myself. I do still look at articles and find good stuff from it, but I've replaced it with paid courses that offer hands-on examples.
This isn't true. I put in a great deal of effort in college and struggled to learn. After college I changed the way I interacted with information, and found that I could learn and remember orders of magnitude better by using studying and practice techniques that mapped more closely with how I thought about information.
> Every learning method you can think of has been thought of before and all variations have been implemented in classrooms throughout time. It is mostly pseudo-science.
This is wrong. Not every "learning method" is pseudo-science, neither is comparison of the efficacy of different learning methods. As a trivial example, flat lecture and individual textbook reading is inferior to one-on-one discussion and tutoring with a native speaker if the aim is to learn a foreign language.
I'm not saying it's a learning method. And I don't see how anyone could mistake this for science, so why would it be pseudoscience? It's not really about effort either.
It's just a trick that helps me pay attention in lectures, which a lot of people struggle with. Certainly you have to put the work outside of the classroom as well.
One of OpenAI's founding team members developed Adam [0] well before it was flashy and profitable. It's not like nobody is out there trying to develop new algorithms.
The reality is that there are some great, mature solvers out there that work well enough for most cases. And while it might be possible to eke out more performance in specific problems, it would be very hard to beat existing solvers in general.
Theoretical developments like this, while interesting on their own, don't really contribute much to day-to-day users of linear programming. A lot of smart people have worked very hard to "optimize the optimizers" from a practical standpoint.
No one thought that theorems in number theory would ever be useful but those theorems are now the foundations of tools like wireguard. Computing the next frame of a snowboarding video is much less valuable than improvements in optimization algorithms that are used daily for optimal transport logistics & energy grid optimization. The promise of AI was solutions to practical problems but what we are getting are frivolous cartoons & 6 second "movie" clips.
There is indeed a lot of crossover, and a lot of neural networks can be written in a state space form. The optimal control problem should be equivalent to training the weights, as you mention.
However, from what I have seen, this isn't really a useful way of reframing the problem. The optimal control problem is at least as hard, if not harder, than the original problem of training the neural network, and the latter has mature and performant software for doing it efficiently. That's not to say there isn't good software for optimal control, but it's a more general problem and therefore off-the-shelf solvers can't leverage the network structure very well.
Some researchers have made interesting theoretical connections like in neural ODEs, but even there the practicality is limited.
Very cool. Analysis I was the first "real" math textbook that I (an engineer, not a mathematician) felt like I could completely follow and work through, after a few attempts to get through others like Rudin. Hopefully the Lean companion will help make it even more accessible to people who are familiar with math and programming and looking to learn things rigorously.
Have you heard of JIT libraries like numba (https://github.com/numba/numba)? It doesn't work for all python code, but can be helpful for the type of function you gave as an example. There's no need to rewrite anything, just add a decorator to the function. I don't really know how performance compares to C, for example.
Compared to Matlab (and to some extent Julia), my complaints about numpy are summed up in these two paragraphs:
> Some functions have axes arguments. Some have different versions with different names. Some have Conventions. Some have Conventions and axes arguments. And some don’t provide any vectorized version at all.
> But the biggest flaw of NumPy is this: Say you create a function that solves some problem with arrays of some given shape. Now, how do you apply it to particular dimensions of some larger arrays? The answer is: You re-write your function from scratch in a much more complex way. The basic principle of programming is abstraction—solving simple problems and then using the solutions as building blocks for more complex problems. NumPy doesn’t let you do that.
Usually when I write Matlab code, the vectorized version just works, and if there are any changes needed, they're pretty minor and intuitive. With numpy I feel like I have to look up the documentation for every single function, transposing and reshaping the array into whatever shape that particular function expects. It's not very consistent.
Yeah, in case anyone else has the misfortune of having to work with multi-dimensional data in MATLAB, I'd recommend the Tensor Toolbox, Tensorlab, or the N-way Toolbox.
There are certainly some aspects of it that are inelegant, in the interests of backwards compatibility, but otherwise I don't know what you are talking about. Matlab supports >2d arrays just fine, and has for at least 20 years.
> Now, how do you apply it to particular dimensions of some larger arrays?
What exactly is the complaint here? If I write a function on a 2x2 array, there's no way to apply it to tiles in a 3x2x2 array? But there is - take a slice and squeeze. What's the issue?
Edit: if the issue is some weird version of "why can't I apply my function to an arbitrary set of indices" (like the 0,1 in a 2x2x2 array) I'm at a loss for words (because that's a completely different shape - indeed it's the entire array).
This looks great! I've been needing something like this for a while, for a project which is quite compute-heavy and uses lots of threads and recursion. I've been using valgrind to profile small test examples, but that's definitely the nuclear option since it slows down the execution so much. I'm going to try this out right away.
That's a good observation, and it is indeed true for many Markov chains. But your counterexample of the identity matrix is not quite right; every vector is an eigenvector of the identity, so there is no "realignment" needed.
More generally speaking, you're asking when the iteration `x_+ = Ax` converges to a fixed point which is an eigenvector of A. This can happen a few different ways. The obvious way is that A has an eigenvector `v` with eigenvalue 1, and all other eigenvalues with magnitude < 1. Then those other components will die out with repeated application of A, leaving only `v` in the limit.
For Markov chains, we can get this exact property from the Perron-Frobenius theorem, which applies to non-negative irreducible matrices. Irreducible means that the transition graph of the Markov chain is strongly connected. If that's the case, then there is a unique eigenvector called the stationary distribution (with eigenvalue 1), and all initial conditions will converge to it.
In case A is not irreducible, you may have different connected components, and the stationary distribution may depend on which component your initial condition is in. Going back to the n x n identity matrix, it has n connected components (it's a completely disconnected graph with all the self-transition probabilities = 1). So every initial condition is stationary, because you can't change anything after the initial step.
If you're interested to learn more about aerodynamics I would highly suggest learning a bit of classical aerodynamics. It will not be software oriented, since most of the theory deals with approximating very complicated behavior with simple analytical models.
It could be interesting to do a comparison with finite volume methods to see when/how those approximations break down.
Totally newbie question - 'approximating very complicated behavior' - this seems like a perfect problem for ML to me. Is this something that's used or explored ?
It's absolutely being explored. There is a lot of active research into using ML to learn solutions of PDEs (Navier-Stokes in this case). It's not my field so I don't know much about the specifics.
The works that I've read train an NN on numerical solutions for different geometries and boundary conditions. Then they try to infer the solutions for configurations outside the training set, which should be much faster than recomputing the numerical solution.
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.