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Homology
- Avicebron 7y agoCool. I was looking at software for protein homology minutes before I refreshed HN...small world
- chas 7y agoThis sort of homology[1] or this sort[2]? It looks like this is about the first one and proteins usually involve the second one. If they are more related than I expect, I would love to find out. [1] https://en.wikipedia.org/wiki/Homology_(mathematics) https://en.wikipedia.org/wiki/Homology_(mathematics) [2] https://en.wikipedia.org/wiki/Sequence_homology https://en.wikipedia.org/wiki/Sequence_homology
- gjm11 7y agoIt 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.)
- mlevental 7y agohow did you draw the diagrams?
- bmer 7y agoHow is an open curve "with boundary", but a closed curve "without boundary"?
- knzhou 7y agoA closed curve could be the boundary of something, but it has no boundary itself. The boundary of an open curve is its endpoints, a closed curve has no endpoints. It's important to distinguish having a boundary vs. being the boundary of something, since in some sense the difference between the two is the whole point of homology.
- cka 7y agoA curve is a 1-dimensional space. Even though you draw a picture of it in a 2 (or maybe higher?) dimensional space, you should think of the set of points in the curve as being the only points that we care about. In a closed curve, you can never fall out of the space by moving around in it. In a curve with endpoints, you can fall out of the space by walking across one of the endpoints, so the endpoints are considered to be boundary components.
- sweeneyrod 7y agoThat'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 7y ago>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.
- enchiridion 7y agoThis seems like a good place to share a gentle intro to persistent homology I recently read. https://towardsdatascience.com/persistent-homology-with-examples-1974d4b9c3d0 https://towardsdatascience.com/persistent-homology-with-exam...
- Archit3ch 7y agoDoes this apply to electromagnetism?
- at_a_remove 7y ago"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.
- outlace 7y agoIn 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
- ternaryoperator 7y agoCould someone explain whether this has any practical applications and, if so, what they are?
- lidHanteyk 7y agoHere 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_gm4hWLvFtJX0WUueVO65uhvBPrA https://www.youtube.com/playlist?list=PLSekr_gm4hWLvFtJX0WUu... 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.
- throwlaplace 7y agothis 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
- heinrichhartman 7y agoHomology (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?
- Koshkin 7y agoIn 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/