
A Visual Introduction to Morse Theory - Topolomancer
http://bastian.rieck.me/blog/posts/2019/morse_theory/
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tshanmu
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.

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Topolomancer
Thank you very much for this kind feedback!

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enriquto
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)

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llamaz
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.

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Topolomancer
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.

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llamaz
I just sent a message to the email on your website :)

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krick
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.

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lixtra
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)

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krick
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.

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lixtra
> 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.

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krick
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.

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lixtra
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.

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

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Topolomancer
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?

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timtaco
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

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dosshell
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.

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Topolomancer
The confusion was unintentional, but this is why I added a footnote in the
article :)

