
Homology - mathgenius
https://www.tungsteno.io/post/exp-homology/
======
outlace
In case anyone is interested in delving deeper, I wrote a detailed series of
posts on persistent homology including the fundamental math and getting a
working python example:
[http://outlace.com/TDApart1.html](http://outlace.com/TDApart1.html)

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enchiridion
This seems like a good place to share a gentle intro to persistent homology I
recently read.

[https://towardsdatascience.com/persistent-homology-with-
exam...](https://towardsdatascience.com/persistent-homology-with-
examples-1974d4b9c3d0)

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heinrichhartman
Homology (and Cohomology) are one of the key methodical innovations in pure
Mathematics in the 20th century. It started as the topological concept
described in the article, but the core idas apply much more general. Todays:
Algebraic Geometry, Number Theory, Category Theory, Singularity Theory, String
Theory, etc. are full of "cohomology". There are thousands of different
flavours in use, that measure different properties of "spaces" of interest.

Much of category theory was initially created to deal with "complexes of
abelian objects" that are used to define cohomology theories.

Working with cohomology was my bread and butter, when working as a
mathematician. I always hoped to apply sone idas to Computer Science as well,
when I entered the field, but I did not find any sane ways to make use of that
"technology". E.g. homology of graphs is pretty boring, since they are only
1dimensional. Is anyone aware of any CS applications of Homology?

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throwlaplace
this is a pretty discussion but not a very good one if you don't already know
some things.

for example

>the holes of dimension 0 are unconnected components

in this sentence

>the holes of dimension 0 are unconnected components, the holes of dimension 1
may be surrounded by a loop, whereas those of dimension 2 may be enclosed by
surfaces

stands out because it's the only classification that's formal (i actually
don't know what the meaning of unconnected component here is but i suspect it
means a point?).

then C(X) is defined informally as

>The set of "objects of dimension 𝑘" will be denoted 𝐶𝑘(𝑋)

but this informal definition doesn't say explicitly that it's the set of
objects in X (until later).

then orientation is defined in a very confusing way

>for now it will be some additional property that takes two opposite values
("one the negative of the other one")

but i will say that the discussion of quotienting was kind of an aha moment
for me (about why you'd rather work in the quotient than the set itself) and
i've been dipping toes into algebra for a loooooong time.

anyway you kind of have to read it like a poem (in that you should try to get
a feeling rather than a precise idea) but that's not very useful.

also i'm interested in how the diagrams were created. looks like latex + tikz?

also it would've make more sense to show

>im ∂𝑘⊂ker ∂𝑘-1

instead of

>im ∂𝑘+1⊂ker ∂𝑘

to stay with the notation in the previous sentence

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bmer
How is an open curve "with boundary", but a closed curve "without boundary"?

~~~
sweeneyrod
That's what open and curve mean in topology. For example, the set of x with 0
<= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously
confusing terminology, since sets can be both closed and open, or neither...

~~~
rgossiaux
>That's what open and curve mean in topology. For example, the set of x with 0
<= x <= 1 is open, whereas 0 < x < 1 would be closed. It's infamously
confusing terminology, since sets can be both closed and open, or neither...

You have it backwards; [0, 1] is closed and (0, 1) is open. This gives the
terminology "open interval" and "closed interval".

The confusion about "why does a closed curve have no boundary" likely comes
more from the word "boundary". The point is that for an n-dimensional object,
the boundary is (n-1)-dimensional. For a curve, which is 1-dimensional, the
boundary is 0-dimensional, ie points-- so we're looking for endpoints, and a
closed curve doesn't have any.

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at_a_remove
"Euler-Poincaré Characteristic." I had noticed this pattern in shapes along
the various dimensions when I was a kid but none of my math teachers knew what
I was talking about. Never asked about it when I went to college, but I'm glad
that there is a _name_ for it.

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Koshkin
In find this resource one of the best:
[https://jeremykun.com/2013/04/03/homology-theory-a-
primer/](https://jeremykun.com/2013/04/03/homology-theory-a-primer/)

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Avicebron
Cool. I was looking at software for protein homology minutes before I
refreshed HN...small world

~~~
gjm11
It turns out that that's a _completely different_ meaning of "homology".

"Homology" really just means something like "correspondence". So it has, among
its meanings,

1\. the one in the article here: a way of associating topological spaces with
algebraic structures, so that you can study one by studying the other.

2\. the biological one: instances where two (parts or aspects of) living
things are similar on account of common descent.

It's not 100% impossible that the two notions might come into contact with one
another. DNA can become knotted, and there are enzymes called topoisomerases
that enable DNA strands to pass through one another to unknot them. (Maybe
something similar happens with proteins?) Knots are topological objects and
homology is one of the tools mathematicians use to study them. Perhaps one day
it will turn out that particular DNA or protein sequences tend to lead to
particular sorts of knotting that are biologically significant (e.g., maybe
they control the transcription rate of DNA or the shape of proteins), and
perhaps it will turn out that they are selected for, and then perhaps from
time to time biologists will find regions in two organisms' DNA or proteins
that exhibit mathematically similar kinds of knotting because they're
descended from an ancestor that found that kind of knotting useful. In that
case they would have homology in their homology.

(Probably (k)not, though.)

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Archit3ch
Does this apply to electromagnetism?

~~~
Koshkin
Differential forms, de Rham cohomology...

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ternaryoperator
Could someone explain whether this has any practical applications and, if so,
what they are?

~~~
lidHanteyk
Here is a practical tutorial on the theory and design of systems which
integrate many sensors into a single coherent collage:
[https://www.youtube.com/playlist?list=PLSekr_gm4hWLvFtJX0WUu...](https://www.youtube.com/playlist?list=PLSekr_gm4hWLvFtJX0WUueVO65uhvBPrA)

You will need a strong grasp of category theory and topology in order to
tackle this subject, though. The article serves as a good introduction,
although it itself requires a mathematical background.

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mlevental
how did you draw the diagrams?

