Massive thanks for article and more in particular this site. Ever since high school I have struggled with n-d space (for n>3), this site and previous article in the blog http://bastian.rieck.me/blog/posts/2019/manifold/ is an amazing introduction.
There is a (tiny) insight that I find missing in this introduction, the fact that the complexity can lie either on the function or on the manifold. The two extreme--and most common--cases of Morse theory are: a very complicated manifold with a simple function on it (e.g. one coordinate), or a very complicated function defined on a flat manifold. The theory applies the same to both cases but the insights are very different.
May I add two beautiful texts of historical interest but amazingly readable:
Thanks for this! I'm doing an undergrad thesis applying TDA to EEG.
Currently I've been trying out using a sublevel set filtration on EEG data using a sliding window. For some reason the total persistence (sum of barcode lengths) can classify seizure vs non-seizure time segments.
Do you have any idea why it's able to do this, or where I can learn more?
I know that seizure events correspond to less determinism, as opposed to more chaos.
That sounds super interesting! I know that there's a lot of work by Perea and Harer about topological time series analysis out there. Most of it is based on sliding windows to my understanding. I would say that the total persistence is a quantifier of the topological complexity of an object (at least when being evaluated in relation to something else; obviously, we can make it as a large as we want by just scaling our weights accordingly)...
Let's discuss this further offline if you want! Drop me an e-mail and I can rope you into my current research on this topic.
Could please somebody provide some kind of motivation for a non-mathematician on that? I tried to start with reading the previous article, but I have no idea, what I should be doing to find myself in need of the tool I'm trying to understand here.
Author here; I am interested in topological data analysis (TDA) for machine learning. Morse theory is one of the tools to analyse manifolds, which are the 'bread and butter' in ML data. I realise that I should have made the connection to TDA more obvious. If you are interested in a general overview, I can recommend this article here:
Morse theory can be used to prove that certain solutions exist/don’t exist.
A simple example is: you have an angular robot arm that can turn 360 degrees. Can you come up with a smooth algorithm that moves it from any current position to any other position? Morse theory answers this because your configuration space is a torus.
Edit: Maybe one could say that Morse theory is the intermediate value [1] theorem for manifolds on steroids.
Not sure if I understand you correctly. Do I need to prove that out of mathematical curiosity or for some profit? I mean, if it's the latter, isn't that kind of obvious (and the solution seems to be an exercise in geometry, not differential topology)? Maybe there are examples, where it would be much harder to see without some kind of sophisticated mathematical tools, but I don't see how it helps me in your example.
> Do I need to prove that out of mathematical curiosity or for some profit?
Knowing that a solution (or how many) exists or not is valuable because it tells you wether it’s worthwhile to search for one. The robot arm example above helped an engineer friend of mine because his boss was not happy with his non-smooth solution. I could provide him with the papers that you cannot do better. This saved him time. Is it worth the overhead to learn Morse theory as an engineer? Probably not. Is it worthwhile to know it exists and ask a mathematician? I would argue yes.
Ok, that makes sense: apparently, I don't understand what is meant by "smooth algorithm". I assumed that it simply means you don't have to stop the manipulator on the path from A to B, but if it's something different, that might be helpful indeed. I'm failing to find a clear definition of what it actually meant when used like that, though.
Say you want to move from A to B on a circle. A simple algorithm would be to just go the shortest way from A to B which is well defined if A and B are not antipodal. If they are, you could define to always go against the clock.
Now this algorithm is fine, but it is not continuous (smooth), because a small variation in an antipodal point gives two vastly different paths.
Morse theory is basically the toy case of a very general way of thinking about Lagrangians / Energy landscapes in physics. You can think of the Morse function as a kind of classical Hamiltonian, paths of steepest descent (gradient descent with respect to the morse function) can be identified as Instantons of supersymmetric quantum mechanics. This was the revolutionary insight of Witten in the 1980s ("Supersymmetry and Morse theory"). Another down to earth interpretation is to think of those paths as elastic rubber bands in high dimensional curved space.
Starting from this insight Witten and others identified sets of other Morse theory like constructions (Donaldson theory got condensed from ~1000s of pages to 40, Witten's treatment of Chern-Simons theory resulted in a fields medal).
> Morse theory is basically the toy case of a very general way of thinking about Lagrangians / Energy landscapes in physics. You can think of the Morse function as a kind of classical Hamiltonian, paths of steepest descent
are you sure that these sentences will be useful to a non-mathematician?
An easier concept to a layman may be the combinatorial properties of topographic maps. For example, the number of mountain peaks, lakes, and mountain passes on an island are not independent but must satisfy a strict numerical identity.
For simple applications, you have not know too much. In order to understand, you have to acquire some mathematical knowledge from related areas (to Morse Theory).
Thanks, I guess that answers the question. I'll have to spend some time reading to actually understand, though. Any reading/video-lecture suggestions that I should go through besides the links in the article?
Author here; I have the same issue with Android, but it seems related to `MathJax`. I initially said that formulas should scale to 100% of the surrounding text...
Can someone with more CSS knowledge chime in here?
Hey, I know Morse code - I was a squad radio operator in the military!
I can tell you that you don't memorize Morse code with math, you memorize it by creating syllabic mnemonics and the key is to learn how to trust your brain when the broadcast is at higher rates.
Of course there is also Chinese Morse code operators that broadcast at such higher rates that we couldn't intercept easily
