The topology of impossible spaces | Hacker News Reader

The topology of impossible spaces

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55 pointsdanharaj3 comments

Cohomology is one of the major ideas of modern mathematics. Not just an abstruse technical device for proving other abstruse technical facts, it's beginning to see use data analysis as well. Cohomological ideas can be found in surprising places. If you liked this article, you might also like this one, which explains how carrying in digit arithmetic is a cocycle: http://www.math.wayne.edu/~isaksen/Expository/carrying.pdf

I consider topology an important branch of mathematics. Morse theory[1] is very powerful to prove existence or non existence of solutions in other spaces of mathematics. I saw it applied to robot motion planning [2] (and could use that to prove to an engineer friend that the sort of algorithm he was looking for cannot exist), so yes it has practical real world applications.

On the other hand I would be surprised to see data scientists apply cohomology in their daily work (citations?).

Also while a short glimpse at your digit arithmetic gives me fun (i’m far from fully getting it that fast) I don’t see any deeper insight we get from it at this stage. I would be careful to propose to someone to learn cohomology if they were interested in data science or arithmetic. Most likely the time could be used in a better way.

[1] https://en.m.wikipedia.org/wiki/Morse_theory [2] https://arxiv.org/abs/math/0111197

(Co)homological methods in data science fall under the umbrella of topological data analysis, which is quite a new approach relative to more standard ones. In particular, persistent (co)homology is the applicable theory here, because it computes homology at varying spatial resolutions.

> so while a short glimpse at your digit arithmetic gives me fun

The fun is the point of it :) It's easy to get discouraged by very abstract, very powerful mathematical ideas, so having fun little toy examples is nice.

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