What are...simplicial complexes?
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- čas přidán 2. 08. 2024
- Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
At 8:11, the one object sets correspond to dimension zero, not one. This always confuses me and I messed that up, sorry! That is also why I messed up the notation on the triangle - one should better start counting at zero because of this "of-by-one-error".
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-a...
Simplicial complexes.
en.wikipedia.org/wiki/Simplic...
Abstract simplicial complex.
en.wikipedia.org/wiki/Abstrac...
community.wolfram.com/groups/...
Pictures used.
en.wikipedia.org/wiki/Simplic...
quantdare.com/understanding-t...
www.cs.columbia.edu/~suman/av...
www.routledgehandbooks.com/do...
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/...
Always useful.
en.wikipedia.org/wiki/Counter...
Mathematica.
demonstrations.wolfram.com/Si...
#algebraictopology
#topology
#mathematics
This explanation was exactly what i was looking for when i was stuck in my script. Thanks 😊
Thanks for the feedback; I am glad that the video was helpful! I hope you enjoy your topology journey!
Thank you for your lecture that is very helpful for me to understand how to form the basic energy of universe.
I am glad that the video was helpful, thank you so much for the feedback. I hope you will enjoy AT!
Awsome presentation as always
Thanks again!
Let me know how you like homology. It is awesome, but a bit unintuitive. So there is probably not a canonical way to explain it. (Not that there is for any other topic, but certainly one sometimes has "more canonical" ways, whatever that means.)
great explanation thanks!
You're welcome! I hope you enjoy your AT journey!
thank you! helped me understand a lot
I am glad that the video was helpful. Thanks for the feedback!
Thanks for adding complexity , I would agree.
Thanks for the feedback! I hope the video was helpful and enjoyable.
Thank you!
Thanks you are welcome, I hope you enjoyed the video!
Great video thanks!!
Glad that you liked it, you are welcome! I hope you will enjoy AT as much as I do ;-)
You are amazing. Thank you.
I am learning about Topological Data Analysis algorithms.
I do not deserve your praise: I try to explain things they way it works best for me. Really very selfish ;-)
Nevertheless, I hope you enjoyed the video, and it will help you with topological data analysis (which is one of the coolest fields of modern topology).
"Hatcher’s book (I sometimes steal some pictures from there)." 😂😂😂 hatcher the homology god 💜
Indeed ;-)
I really appreciate this video and how you try to make it simple! I am trying to understand what the 4 dimensional simplicial complex would look like :)
I liked the two definitions of a simplicial complex "smallest convex set with n vertices" and that "any subset of points of a simplicial complex also makes a simplicial complex". I have been thinking, it seems that each simplicial complex is made up of a couple of the n-1 simplicial complex in different basis glued together.
I am a bit confused by what you meant at time 6:50 when you talked about gluing c and d together and what it means for an intersection to be a "face of both" simpliical complexes?
You are trying to visualize 4D? Oh, you are brave ;-) There are essentially three ways trying to do this, say for the 4D tetrahedron which is the prototypical 4D simplicial complex:
First, via a projection. Maybe you have seen the picture of the 4D cube? See the animation here
en.wikipedia.org/wiki/Tesseract
This is the most popular form to illustrate 4D, but I am not sure whether that works for me. I still find that pretty much mysterious.
Second, by adding a non-space dimension, e.g. via a movie where time is the fourth dimension.
www.dtubbenhauer.com/slides/my-favorite-theorems/48-volume-balls.pdf
www.researchgate.net/publication/342697434/figure/fig1/AS:960092123975683@1605915208302/Examples-of-mathematical-drawings-to-communicate-geometry-and-topology-a-Foxs.png
Not a new idea, but it is still hard to find good illustrations that are freely available. Not sure why this is not well-known. I am guessing right now that during the history of computer-animation someone decided that projections are nicer, and then everyone just copied that. That what humans do, I guess. So most pictures you find are projections, but at least for me movies work much better. To illustrate the 4D tetrahedron is a fun exercise ;-)
Finally, abstractly. Here is the tetrahedron: {[0]}, {[0],[1],[0,1]}, {[0],[1],[2],[0,1],[1,2],[0,2],[0,1,2]} etc. The last one is the triangle with [0,1,2] being the face, [0,1] being the edge from 0 to 1 etc. The 4D tetrahedron now has [0,1,2,3,4] and all of its “subsimplices”. This of course is not very visual, but easily works in any dimension (or is what your computer would use).
Oh, that was a lot of waffle! Sorry for that. To come to your question, proably the confusion comes from what “face” means. This is 3D terminology, so it should be the (n-1)-dimensional part of an n-simplex. Let me illustrate that using the abstract notation, which is the third above. Let us do a small example. Say you have two “volumes” [a,b,c,d] and [e,f,g,h], and they intersect. The intersection should be a face of both, e.g. [a,b,c]=[e,f,g] if a=e, b=f, and c=g. Does that make sense?
@@VisualMath I see what you mean, basically the simplex has to look nice :). Only part of an edge can't be shared or there can't be random simplexes going through other simplexes.
I thought perhaps c + d meant you were adding the two pictures together in some way haha! I think it would have been better if you added a comma like "c, d" or do "c and d".
Just curious, how do we call the addition of two n simplexes?
I think the 4D hypercube is the explanation that has made the most sense to me. I feel like if you connected a 2D hypercube to a another 2D hypercube with aluminum wires the length of a side, they would automatically go into the 3rd dimension. There's no way do to that for a 3D hypercube, unless... Watch out for my research paper!!! XD
@@kaushikdr I see - now I understand the question ;-) Yes, the “+” should be an “and”, which always worked for me but maybe not for everyone. Thanks for the tip - I try to be more careful in the future.
Not sure what addition of n-simplexes actually is; a priori you can't add them. I guess it doesn't have a name
Can I have the software for visualizing simplicial complex from point cloud data?
Sorry, I am unsure what you mean. What is it you want, maybe you can elaborate and I will see what can be done? I think I never used any fancy visualization techniques in this video.
How can we draw Simplicial Complexes using Python or Matlab?
That is a good question - I would really like to know that as well!
Applied topology has quite a few things one can do, but they sometimes go under a different header such as persistent homology. See e.g.
github.com/appliedtopology/javaplex
Maybe that is helpful?
@@VisualMath Is there any relations between Hypergraphs and Simplicial Complexes?
@@navneetsinha451 Simplicial complexes are special cases of hypergraphs, see for example
www.reddit.com/r/math/comments/8vhjre/difference_between_hypergraph_and_simplical/
@@VisualMath Thanks sir, it was really helpful
@@navneetsinha451 Welcome.
P.S.: Sorry for being annoying, but I go by "they/them" so "sir" is not the correct way of addressing me.
Are you married
Let me not answer this one ;-)
Thank you for this awesome lesson please i need your support as the beginner of this field