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 ?
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.
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.
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'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.
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.
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?
@@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.
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?
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.
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 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?
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.
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.
Every time I've followed a lead about some paper that claims to use persistent homology to study something, that has never led to someone actually using the technique outside of an academic paper. Caveat emptor.
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.
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.
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
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
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.
18:15 this is very important and I think it’s the essence of the concept of simplex. Nice!
At 21:30, I think the sum needs to run from 0 to n, not 1 to n.
He really did just go and say "grow some balls" without laughing. What a lad
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
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.
在这节课中，我们介绍了网格（mesh）和简单复形（simplicial complex）的基本概念。网格是由顶点、边和面构成的组合结构，具有特定的连接性。我们讨论了抽象简单复形与几何简单复形之间的区别，并探讨了它们在离散微分几何中的应用。课程中还介绍了凸集和凸包的定义，简单形的概念，以及如何通过加权组合来描述简单形的坐标。此外，课程还涉及了数据分析中的拓扑数据分析及其在实际应用中的重要性，如持久同调的概念。
亮点:
00:11 我们将介绍组合表面或更常见的网格，这是我们定义其他几何形状的基础。通过讨论简单复形的抽象和几何形式，我们将深入理解网格在离散微分几何中的重要性。
-简单复形是由顶点、边、三角形和其他高维元素组成的集合。我们将讨论抽象简单复形和几何简单复形之间的区别，以及它们在空间中的位置。
-拓扑空间的概念是理解网格与连续数学对象之间关系的关键。拓扑空间强调点之间的连接性，而不涉及具体的几何位置。
-我们将探索凸集的定义，并通过对比不同形状来理解什么是凸集。通过具体示例，我们将判断哪些形状是凸的，哪些不是。
06:03 凸包是包含给定点集的最小凸集，它可以通过将形状紧密包裹来形成。通过交集的方式，我们可以得到这一凸包的形状，确保它不变得非凸。
-通过可视化不同的凸集，我们可以更好地理解交集如何收缩到最小的凸包。这样的图形化表示使得数学概念更加直观，便于学习和应用。
-在几何中，凸集的概念至关重要，因为它帮助我们理解如何定义简单单形。简单单形是基本的几何结构，不同维度的简单单形具有不同的形状，比如点、线段和三角形。
-凸包的概念还可以通过线性独立性来进一步理解。线性独立的向量集在几何中具有重要意义，因为它们可以用于构建更复杂的形状和结构。
12:05 仿射独立性的定义是理解几何形状的重要基础。通过具体的例子，我们可以更好地理解怎样的点集合是仿射独立的，如何用它们构造简单形状，如线段和三角形。
-要判断一组点是否是仿射独立的，可以通过绘制向量来观察它们之间的关系。比如，两个线性独立的向量形成的三点集合是仿射独立的，而线性相关的则不是。
-简单形状（simplex）的定义是基于仿射独立点的集合。k维简单形状是由k+1个仿射独立点的凸包构成，这为我们理解几何形状提供了一个清晰的框架。
-不同数量的仿射独立点形成不同的简单形状。例如，两个仿射独立的点形成线段（1-simplex），三个形成三角形（2-simplex），而四个点在三维空间中形成四面体（3-simplex）。
18:07 我们了解到，单纯形的维度是有限的，在特定维度中只能存在特定的单纯形。例如，二维中只能有二元单纯形，而三维中只能有三元单纯形，无法存在更高维的单纯形。
-在讨论单纯形时，使用重心坐标（barycentric coordinates）可以帮助我们理解单纯形的结构。这些坐标结合了顶点的权重，使得我们能够表示单纯形内的任意点。
-标准n单纯形是所有坐标和为1且非负的点的集合。这种单纯形也被称为概率单纯形，因为概率的特性符合这些条件。
-单纯形复合体（simplicial complex）是由多个单纯形组成的集合，且具有特定的交集性质。理解这些性质将有助于我们更好地分析和使用这些单纯形。
24:11 在几何上，简单复形是一个简单体的集合，其中任何两个简单体的交集也是简单体。每个简单体的每个面也属于这个复形，这种结构在拓扑学中具有重要意义。
-空集在简单复形中扮演重要角色，它被视为一个负一维的简单体。这一技术细节虽然有时让人困惑，但它帮助我们理解简单复形的概念。
-简单复形的几何定义强调了空间中元素的位置关系，两个三角形的交集可以是边或顶点。此定义确保了复形的结构维持一致性，避免了不符合条件的组合。
-抽象简单复形的定义则更为广泛，任何集合的所有子集也必须包含在复形中。这种抽象的定义使得我们可以更灵活地处理不同的数学对象，而不局限于几何形状。
30:15 抽象单纯复形为拓扑空间提供了离散的类比，使我们能够理解几何体如何相互连接，而无需担心形状或位置。这种方法在数据分析中变得尤为重要，尤其是在理解复杂数据结构时。
-图是一个重要的抽象单纯复形示例，它由顶点和边组成，代表一维的连接结构。通过这种方式，我们可以直观地理解网络的基本构成和连接关系。
-持久同调是拓扑数据分析中的一个关键概念，允许我们通过观察数据的连接性来识别数据结构。这个方法能够帮助我们在处理复杂数据时提取出有意义的特征。
-随着球体半径的增加，持久同调方法可以揭示数据点之间的连接关系，识别出哪些结构是持久存在的。通过图形化这种连接性，我们可以更好地理解数据的内在结构。
36:19 持久性图的概念涉及如何通过分析数据中连接组件的出现和消失来理解特征的持久性。这些特征的持久性越长，就越可能是真实的，相反，快速出现和消失的特征则可能是噪声。
-持久性同源性在材料科学中的应用可以帮助表征玻璃。研究人员发现，玻璃中存在的强特征在随机排列的原子中并不存在，从而提供了一种新的量化玻璃状态的方法。
-在几何数据分析中，研究人员通过生长球体来获取点的特征签名。这种方法可以在不同的表面上对比特征，从而识别特定点的位置，例如手指的中间与拇指之间的区别。
-持久性同源性还被应用于神经网络的研究，以区分健康的大脑和存在疾病的大脑。通过分析连接信息，研究人员可以更好地理解社区成员之间的关系。
42:24 视频主要讲述了简单复形的星、闭包和链接的概念。这些概念有助于理解与顶点相关的几何结构，以及如何通过这些操作来描述复形的性质。
-星的概念指的是与特定顶点相连的所有单纯形。通过这个概念，我们能够识别与顶点相邻的三角形和边，从而更好地理解复形的局部结构。
-闭包操作可以帮助我们获得包含所有相关三角形和边的完整集合。通过从闭包中减去星的内部，我们可以提取出复形的边界，从而更深入地分析其几何特性。
-在讨论方向性时，强调了在几何计算中的重要性。方向性不仅影响边的表示方式，还对后续的积分计算有重大影响，这在离散外微积分中尤为重要。
48:29 有向单纯形可以通过有序三元组来定义，并且在圆形变换下的元组是等价的。随着维度的增加，我们可以用类似的方式定义k单纯形的方向性，涉及偶数和奇数排列的概念。
-当我们讨论有向单纯形时，可以将其视为有序三元组的等价类，两个元组在圆形变换下是等价的。这表明在处理多维单纯形时，方向是可以通过这些元组来统一描述的。
-在定义k单纯形的方向性时，我们考虑偶数和奇数排列的影响。偶排列被认为是正向的，而奇排列则是负向的，这对于理解高维单纯形的性质至关重要。
-有向单纯复形的概念是将每个单纯形赋予一个方向。无论是顺时针还是逆时针的排列，单纯复形中的每个元素都需要明确标识其方向，以便于进一步的几何分析。
54:32 我们可以通过比较简单体的相对方向来理解其几何特性。通过观察相交的最大面，我们可以确定两个简单体是否具有相同的相对方向，这对拓扑的理解至关重要。
-在比较两个三角形的相对方向时，如果它们的交集边缘方向相反，就会被认为具有相同的相对方向。这种关系通过观察简单体的最大面来明确，帮助我们理解几何形状之间的关系。
-在讨论方向时，我们需要考虑不同的几何形状，例如圆柱体与莫比乌斯带。这些形状的三角剖分会影响我们赋予它们的方向性，从而影响我们对其拓扑结构的理解。
-相对方向的概念不仅影响局部几何特性，还能揭示全局拓扑信息。通过了解方向性，我们可以更深入地分析形状的性质以及它们在更高维空间中的表现。
由sider.ai生成
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.
7:48 The convex hull of the two point set in R3 shouldn't be just a line segment connecting the two points instead of the whole cube?
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'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
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!
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.
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?
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.
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.
is the drawing for closure correct? closure 41:35
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?
Like always, awesome material.
Ps 32:00 I think there is an extra }
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.
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
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 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. :-)
errata: 21:58 shouldn't the sum of x_i run from i =0 ?
Hi professor, is DDG just another name of computational geometry? If not, what’s the difference? Thank you.
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 :)
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.
It should be pronounced morbius. It's from the movie morbius. One of the most successful movies in this generation
Dear Prof. Crane, i still don't know what is the mesh. Is it simplicial surface? If yes, is it always is simplicial surface?
Excellent!!
Every time I've followed a lead about some paper that claims to use persistent homology to study something, that has never led to someone actually using the technique outside of an academic paper. Caveat emptor.
Very interesting. Thank you :)
riemannian geometry notes please
great
what the mesh?!
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.
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.