I think this might be the most genuinely, *generally* useful mathematical knowledge condensed into the shortest amount of time that I have ever come across. It's just...overwhelming.
Hello professor Keenan Crane, thank you very much for these pretty useful lectures. I have a question which is not a scientific one. What drawing tool did you use to draw the nice surfaces such as the two shaded surfaces at the upper- left corner of the slide 46:08 ?
This is a great lecture. Learning so much from these videos. I have one question about affine independence. I was thinking if it could also be called as "relative" or "positional" independence since it depends on the location of points instead of an absolute position (origin). Let me know.
21:42 this concept is beautiful! I think its usefulness in probability is not just that all of the coordinates of any point in the standard simplex is between 0-1, but much more importantly, is its use in the whole probability concept due to the nice attribute that the sum of the coordinates of any point is always 1. But I think I have a question. What’s the k for this standard k-simplex? 2?
16:58 question: in the case of 1-simplex, where there are two points (vertices), from intuition I know if these two vertices coincide, they are not affinely independent. The problem is, how can we say so based on the definition of affine independence? It seems like the two coincidental points are also affinely independent because this is already a degraded case where there is only one vector to work with, and their coincidence does not make any difference in this regard.
Thank you very much for your lecture! I haven't understood correctly how persistent homology works and how to build the persistent diagram. I'm wondering may you give some extra references to materials to understand it?
A quick question, does 2-simplex contain vertices and line segments of the triangle (i.e. its faces)? It seems so from the convex hull definition. But we have simplices doesn't contain its faces when talking about star and closure. Thank you!
My question pertains to understanding k-simplicies from a geometric point of view. In particular, what are the dimension"s" a simplex can exist in? 1. Geometrically is natural to think 1-simplex lies in R^2. A) But can a 1-simplex exist in R^1? I think the answer to this is probably a yes. For example: A linearly independent vector OR two affinely independent points (p1, p2) can exist as: a. p1: {0} and p2: {1}. Here both points can be thought of as numbers lying on the number line. b. p2: {0,0} and {1, 1}. Here the points can be thought of a line y=x , where 0
In general, you can properly draw a k-simplex in any Euclidean n-space with n>k. Where you were confused about is the fact that k+1 dimensions are required minimally for the existence of a k-simplex.
11:28 question: what’s the relationship between linear independence and orthogonality? Obviously the latter fits in the former but not vice versa. But is there anything deeper between the two?
This was done in just a few lines of Mathematica-though my solution is pretty lazy: I build a dense (!) adjacency matrix by looping over all pairs of points, then hand this matrix to Mathematica's graph plotting function (overlaid with balls around each point). A better way to do it-especially if you want to draw bigger cliques-is probably to dump the data from a "real" software package for topological data analysis, and plot it in a tool like Mathematica, matplotlib, etc.
For an abstract K complex, the star of vertex i, is the collection of all simplices σ ∈ K such that i ∈ σ. If that is the case why are the "outer edges" not considered to be in the star as well?
@@emrekara4896 A face is a simplex formed by a subset of vertices of your original simplex, not any subset of it. E.g. I can draw a mouse inside a large triangle and it won't be a face, while being a subset. I think indeed it should require the intersection to be a face of each of them, otherwise e.g. two intersecting segments in R2 together with the point of their intersection would be a complex, while it seems reasonable to require the four segments (resulting from the intersection) to be present in the complex as well.
Dear Prof. Crane. I'm really appreciate to your lectures :) I have a question. In 9:20, you said that the convex hull of S={(1,1,1),(-1,-1,-1)} in R^3 is the (unit)-cube. But, I think that the set of line segment connecting two points (1,1,1) and (-1,-1,-1) is the smallest convex set containing S, so that the line segment is the convex hull of S. Where is the missing link?
the set S is all combinations of (1,1,1) so it contains (1,1,-1), (-1,1,-1) , (1,-1,1) etc.. So hence the line segment wont work in your case. I presume this is the correct answer to your question.
Observations: - A k-simplex has $2^{k + 1}$ faces. (23:04) - The possible permutations of orientations of a k-simplex are $(k + 1)!$. (51:17) - An orientation of a k-simplex $A$ may be considered to be negative the orientation of another k-simplex $B$ if the number of swaps of 0-simplices in the tuple to get from $A$ to $B$ is odd. (51:17)
I think professor Crane meant that all combinations of the points ({+-1, +-1, +-1)} and not just (1,1,1) and (-1,-1,-1). However, simply looking at the diagram can be confusing since only two points are highlighted and annotated.
@@vinitsingh5546 I believe your second thought is correct. When I watched this portion the first time, I thought of the default cube in Blender (open source 3d modelling software) which is a 2x2x2 cube centered at the origin.
Just a point of pronunciation at 46:20, since I hear people mispronounce it all the time, Mobius is pronounced "mur-bius", not "moe-bius". The umlaut over the "o" in German gives an "ur" sound. So "Agust Mobius" is "agust mur-bius" and "Kurt Godel" is "kurt gur-del" (not "goe-del"). Both mathematician's names are pronounced with an "ur" sound.
@@ChrisOffner I'm not trolling, nor have I seen what you are referring to. If I am mistaken, I apologize. How should it be pronounced? The pronunciation is entirely miniscule relative to this wonderful lecture. The content is what is most important.
To be fair, this IS a math course. If all the definitions fly over your head, don't worry too much. Focus on building your intuitions first. And I haven't seen any other course that even comes close to the level of clarity in Prof. Crane's lectures. As long as you don't plan to become a pure mathematician, good intuitions should be good enough in most cases.
By way of protesting all commercials on youtube: if i ever meet someone who uses grammarly, they're immediately fired. because they're too stupid to do whatever it is they're doing for a job, that has writing as one of the tasks.
If I ever meet someone who immediately fires someone for using a tool, they're immediately fired, because they're too stupid to do whatever it is they're doing for a job, that has firing as one of the tasks.
0:21 Today: What is a "Mesh?"
1:54 Connection to Differential Geometry
3:02 Convex Set
6:00 Convex Hull
7:34 Example
9:25 Simplex
10:19 Linear Independence
11:38 Affine Independence
13:49 Simplex - Geometric Definition
18:28 Barycentric Coordinates - 1-Simplex
19:27 Barycentric Coordinates - k-Simplex
20:50 Simplex - Example
22:02 Simplicial Complex
23:04 Face of a Simplex
25:03 Simplicial Complex - Geometric Definition
27:00 Simplicial Complex - Example
28:50 Abstract Simplicial Complex
30:34 Abstract Simplicial Complex - Graphs
31:01 Abstract Simplicial Complex - Example
33:10 Application: Topological Data Analysis
38:04 Example: Material Characterization via Persistence
39:23 Persistent Homology - More Applications
41:19 Anatomy of Simplicial Complex
44:23 Vertices, Edges, and Faces
45:23 Oriented Simplicial Complex
45:38 Orientation - Visualized
46:52 Orientation of 1-Simplex
47:58 Orientation of 2-Simplex
49:17 Oriented k-simple
50:36 Oriented O-Simplex?
51:17 Orientation of 3-Simplex
52:25 Oriented Simplicial Complex
54:54 Relative Orientation
I think this might be the most genuinely, *generally* useful mathematical knowledge condensed into the shortest amount of time that I have ever come across. It's just...overwhelming.
This is like the best education channel here.
linear Independence 10:22
affine Independence 11:38
k-simplex 14:06
Barycentric Coordinate, convex combination 18:30
standard n-simplex, probability-simplex 20:50
face of a simplex 23:07
simplicial complex 25:03
abstract simplicial complex 28:51
persistent homology 34:39
closure 41:35
star 42:22
link 43:02
oriented 1-simplex 46:55
oriented 2-simplex 48:00
oriented k-simplex 49:20
oriented 0-simplex 50:39
oriented simplicial complex 52:26
relative orientation 55:00
18:15 this is very important and I think it’s the essence of the concept of simplex. Nice!
This is so high quality that for it to be free and for us to be listening to this for no price seems like stealing!
He really did just go and say "grow some balls" without laughing. What a lad
Excellent!!
Very interesting. Thank you :)
Hello professor Keenan Crane, thank you very much for these pretty useful lectures. I have a question which is not a scientific one. What drawing tool did you use to draw the nice surfaces such as the two shaded surfaces at the upper- left corner of the slide 46:08 ?
Thanks Ahmad. I give some answers to this question at the bottom of this FAQ: keenan.is/questionable
This is a great lecture. Learning so much from these videos. I have one question about affine independence. I was thinking if it could also be called as "relative" or "positional" independence since it depends on the location of points instead of an absolute position (origin). Let me know.
21:42 this concept is beautiful! I think its usefulness in probability is not just that all of the coordinates of any point in the standard simplex is between 0-1, but much more importantly, is its use in the whole probability concept due to the nice attribute that the sum of the coordinates of any point is always 1.
But I think I have a question. What’s the k for this standard k-simplex? 2?
16:58 question: in the case of 1-simplex, where there are two points (vertices), from intuition I know if these two vertices coincide, they are not affinely independent. The problem is, how can we say so based on the definition of affine independence? It seems like the two coincidental points are also affinely independent because this is already a degraded case where there is only one vector to work with, and their coincidence does not make any difference in this regard.
Thank you very much for your lecture! I haven't understood correctly how persistent homology works and how to build the persistent diagram. I'm wondering may you give some extra references to materials to understand it?
At 21:30, I think the sum needs to run from 0 to n, not 1 to n.
thanks for the great lecture, there is an index error in the formula for the standard simplex in case someone is wondering !
A quick question, does 2-simplex contain vertices and line segments of the triangle (i.e. its faces)? It seems so from the convex hull definition. But we have simplices doesn't contain its faces when talking about star and closure. Thank you!
For the slide about abstract simplicial complex, do you mean that a set of size k+1 is called an (abstract) k-simplex? (but not just simplex)
Yes, exactly.
My question pertains to understanding k-simplicies from a geometric point of view.
In particular, what are the dimension"s" a simplex can exist in?
1. Geometrically is natural to think 1-simplex lies in R^2.
A) But can a 1-simplex exist in R^1?
I think the answer to this is probably a yes.
For example:
A linearly independent vector OR two affinely independent points (p1, p2) can exist as:
a. p1: {0} and p2: {1}. Here both points can be thought of as numbers lying on the number line.
b. p2: {0,0} and {1, 1}. Here the points can be thought of a line y=x , where 0
In general, you can properly draw a k-simplex in any Euclidean n-space with n>k. Where you were confused about is the fact that k+1 dimensions are required minimally for the existence of a k-simplex.
11:28 question: what’s the relationship between linear independence and orthogonality? Obviously the latter fits in the former but not vice versa. But is there anything deeper between the two?
Like always, awesome material.
Ps 32:00 I think there is an extra }
Hi, Prof. Keenan, I'm interested in the drawing the filtration complexes at 35:50 . How did you implement it? Is there any existing tool for it?
This was done in just a few lines of Mathematica-though my solution is pretty lazy: I build a dense (!) adjacency matrix by looping over all pairs of points, then hand this matrix to Mathematica's graph plotting function (overlaid with balls around each point). A better way to do it-especially if you want to draw bigger cliques-is probably to dump the data from a "real" software package for topological data analysis, and plot it in a tool like Mathematica, matplotlib, etc.
@@keenancrane Thanks a lot for your useful guideline. I'll try it :)
errata: 21:58 shouldn't the sum of x_i run from i =0 ?
For an abstract K complex, the
star of vertex i, is the collection of all simplices σ ∈ K such that i ∈ σ. If that is the case why are the "outer edges" not considered to be in the star as well?
Because they don't contain i. :-)
About condition 1 in the defination of a simplicial complex，does it require the intersection of two simplices be a face of each of them？
A subset is a face by definition. That goes for the intersection too.
@@emrekara4896 A face is a simplex formed by a subset of vertices of your original simplex, not any subset of it. E.g. I can draw a mouse inside a large triangle and it won't be a face, while being a subset.
I think indeed it should require the intersection to be a face of each of them, otherwise e.g. two intersecting segments in R2 together with the point of their intersection would be a complex, while it seems reasonable to require the four segments (resulting from the intersection) to be present in the complex as well.
He said not to think too much about the empty set abstraction but my conclusion was that 0=ap+bp+cp, must be somewhere so you need it
Hi professor, is DDG just another name of computational geometry? If not, what’s the difference? Thank you.
Dear Prof. Crane. I'm really appreciate to your lectures :) I have a question. In 9:20, you said that the convex hull of S={(1,1,1),(-1,-1,-1)} in R^3 is the (unit)-cube. But, I think that the set of line segment connecting two points (1,1,1) and (-1,-1,-1) is the smallest convex set containing S, so that the line segment is the convex hull of S. Where is the missing link?
the set S is all combinations of (1,1,1) so it contains (1,1,-1), (-1,1,-1) , (1,-1,1) etc.. So hence the line segment wont work in your case. I presume this is the correct answer to your question.
Yeah the issue is yousdefined S differently
Observations:
- A k-simplex has $2^{k + 1}$ faces. (23:04)
- The possible permutations of orientations of a k-simplex are $(k + 1)!$. (51:17)
- An orientation of a k-simplex $A$ may be considered to be negative the orientation of another k-simplex $B$ if the number of swaps of 0-simplices in the tuple to get from $A$ to $B$ is odd. (51:17)
Dear Prof. Crane, i still don't know what is the mesh. Is it simplicial surface? If yes, is it always is simplicial surface?
I am a little confused as to why the convex hull of two collinear points is a Cube and not a line segment in R^3? 8:12
I think professor Crane meant that all combinations of the points ({+-1, +-1, +-1)} and not just (1,1,1) and (-1,-1,-1). However, simply looking at the diagram can be confusing since only two points are highlighted and annotated.
@@vinitsingh5546 I believe your second thought is correct. When I watched this portion the first time, I thought of the default cube in Blender (open source 3d modelling software) which is a 2x2x2 cube centered at the origin.
great
riemannian geometry notes please
what the mesh?!
Just a point of pronunciation at 46:20, since I hear people mispronounce it all the time, Mobius is pronounced "mur-bius", not "moe-bius". The umlaut over the "o" in German gives an "ur" sound. So "Agust Mobius" is "agust mur-bius" and "Kurt Godel" is "kurt gur-del" (not "goe-del"). Both mathematician's names are pronounced with an "ur" sound.
@@ChrisOffner I'm not trolling, nor have I seen what you are referring to. If I am mistaken, I apologize. How should it be pronounced?
The pronunciation is entirely miniscule relative to this wonderful lecture. The content is what is most important.
This one is a bit too abstract...
To be fair, this IS a math course. If all the definitions fly over your head, don't worry too much. Focus on building your intuitions first. And I haven't seen any other course that even comes close to the level of clarity in Prof. Crane's lectures.
As long as you don't plan to become a pure mathematician, good intuitions should be good enough in most cases.
By way of protesting all commercials on youtube:
if i ever meet someone who uses grammarly, they're immediately fired. because they're too stupid to do whatever it is they're doing for a job, that has writing as one of the tasks.
If I ever meet someone who immediately fires someone for using a tool, they're immediately fired, because they're too stupid to do whatever it is they're doing for a job, that has firing as one of the tasks.