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A Visual Introduction to Morse Theory

106 points by Topolomancer 7y ago

llamaz 7y ago
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.
- Topolomancer 7y ago
  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.
  - llamaz 7y ago
    I just sent a message to the email on your website :)
mbeex 7y ago
Rendered formulas are too small (Chrome and FF on Windows).
- Topolomancer 7y ago
  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?
krick 7y ago
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.
- lixtra 7y ago
  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. [1] https://en.m.wikipedia.org/wiki/Intermediate_value_theorem https://en.m.wikipedia.org/wiki/Intermediate_value_theorem
  - breadandcrumbel 7y ago
    So basically morse theory can help you to make faster validation conclusions? Does developers/engineers use it?
  - krick 7y ago
    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.
    - lixtra 7y ago
      > 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.
tshanmu 7y ago
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/ http://bastian.rieck.me/blog/posts/2019/manifold/ is an amazing introduction.
- Topolomancer 7y ago
  Thank you very much for this kind feedback!
enriquto 7y ago
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: * J.C. Maxwell, "On Hills and Dales", 1870. https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf * A. Cayley, "On contour and slope lines", 1859. http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf "http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf
timtaco 7y ago
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
- dosshell 7y ago
  I think you are a little bit confused. This is about Morse theory (topology), by Marston Morse. The Morse code you are talking about is something else, by Samuel Morse.
  - Topolomancer 7y ago
    The confusion was unintentional, but this is why I added a footnote in the article :)

106 points by Topolomancer 7y ago

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.

Topolomancer 7y ago
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.
- llamaz 7y ago
  I just sent a message to the email on your website :)

mbeex 7y ago

Rendered formulas are too small (Chrome and FF on Windows).

Topolomancer 7y ago
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?

krick 7y ago

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.

lixtra 7y ago
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. [1] https://en.m.wikipedia.org/wiki/Intermediate_value_theorem https://en.m.wikipedia.org/wiki/Intermediate_value_theorem
- breadandcrumbel 7y ago
  So basically morse theory can help you to make faster validation conclusions? Does developers/engineers use it?
- krick 7y ago
  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.
  - lixtra 7y ago
    > 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.

tshanmu 7y ago

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/ http://bastian.rieck.me/blog/posts/2019/manifold/ is an amazing introduction.

Topolomancer 7y ago
Thank you very much for this kind feedback!

enriquto 7y ago

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: * J.C. Maxwell, "On Hills and Dales", 1870. https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/hilldale.pdf * A. Cayley, "On contour and slope lines", 1859. http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf "http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf http://www.maths.ed.ac.uk/~aar/papers/cayleyconslo.pdf

timtaco 7y ago

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

dosshell 7y ago
I think you are a little bit confused. This is about Morse theory (topology), by Marston Morse. The Morse code you are talking about is something else, by Samuel Morse.
- Topolomancer 7y ago
  The confusion was unintentional, but this is why I added a footnote in the article :)