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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, w

by 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.
    - krick 7y ago
      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.
      - lixtra 7y ago
        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.
mbeex 7y ago
Example: https://umap-learn.readthedocs.io/en/latest/how_umap_works.html https://umap-learn.readthedocs.io/en/latest/how_umap_works.h... 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).
- krick 7y ago
  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?
Topolomancer 7y ago
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: https://towardsdatascience.com/from-tda-to-dl-d06f234f51d https://towardsdatascience.com/from-tda-to-dl-d06f234f51d
orbifold 7y ago
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).
- enriquto 7y ago
  > 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.
  - orbifold 7y ago
    It's useful to a theoretical physicist :).

by krick 7y ago

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.
  - krick 7y ago
    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.
    - lixtra 7y ago
      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.

mbeex 7y ago

Example: https://umap-learn.readthedocs.io/en/latest/how_umap_works.html https://umap-learn.readthedocs.io/en/latest/how_umap_works.h... 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).

krick 7y ago
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?

Topolomancer 7y ago

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: https://towardsdatascience.com/from-tda-to-dl-d06f234f51d https://towardsdatascience.com/from-tda-to-dl-d06f234f51d

orbifold 7y ago

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).

enriquto 7y ago
> 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.
- orbifold 7y ago
  It's useful to a theoretical physicist :).