I am having a doubt at 17:30 For example if I consider simplicial complex spanned by 4 vertices v0,v1,v2,v3, and let vivj denote an oriented edge then the chain v0v1+v1v2+v0v2+2*v2v3-2*v0v3 is a cycle as the boundary is zero but I am not able to see the alpha, beta's etc. in this chain that add to zero that you are talking about at 17:30. Please can you explain. Thanks
We will be away for Christmas holiday, but in January we will resume covering chapter 2. We will likely also cover chapter 3 in the Spring, but may not make it through chapter 4.
Thank you so much for the lucid presentation and explanation. Are you going to cover chapter 3 of Hatcher as well ? If you already did, where can I find the video ?
Slowly absorbing the fundamentals of algebraic topology but wondering if it gets expanded from metrics on rings and links on homeological space to all imaginary spaces much as in arithmetic the integers are generalised through fractions to transcendentals and exhaust real number possibilities. Impatient when ploughing through the definitions and their restrictions so wonder if later there is topology using weirder algebras.
The explanation you gave in the beginning about why pi_1 isn't abelian is incorrect and may confuse students. It is clear that a and b^{-1} do not commute because they aren't even loops in the first place, so they don't even represent elements in pi_1. Instead you would have to look at a pair of loops, say ab^{-1} and bc^{-1}, and discuss why they don't commute in pi_1.
Where did my comment go? I really need an explanation of how these delta complexes are groups. The fundamental group I get; it's the set of loops in a given space that are not equivalent. The Cn and delta n groups seem to be sets of vertices, but what does it _mean?_ Are we still talking about cycles? What is the group's combination operation? What is the inverse of a vertex? I don't get it, please help!
The C_n is the free group generated by the n-simplices. In the case of C_0 its just linear combinations of the vertices. In the case of C_1 its linear combinations of edges. In the case of C_2 its linear combinations of fillled in triangles. The linear combinations themselves don't have to mean anything per se. They're just algebraic objects. Formal sums of n-simplicies but what he's trying to communicate to you in this video is that we can think of linear combinations of these simplicies as "paths" in the complex. In the case of C_1 linear combinations of edges corresponded to paths in the graph. Really the linear combinations don't have to be single paths, they can be disjoint paths. Hence the name chains instead. I'm not sure what you mean by delta_n? Assuming your talking about the boundary operator, its not a group, its a group homomorphism from C_n to C_{n-1}. The way its defined you can see its output is indeed a linear combination of (n-1)-simplicies. Hope this helps!
Topological holes cannot be shrunk down to zero -- non null homotopic. The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist. "Always two there are" -- Yoda.
@@mmeister8582 Positive curvature is dual to negative curvature -- Gauss or Riemann curvature. Contravariant is dual to covariant -- vectors, functors or a dual basis in Riemann geometry. Riemann geometry contains a hidden dual basis. Singularities (points) are dual. Black holes are positive curvature singularities and white holes are negative curvature singularities. The big bang is an infinite negative curvature singularity -- dark energy is repulsive gravity (inflation). Gaussian negative curvature is defined using two dual points! Points are dual to lines -- the principle of duality in geometry. Homology is dual to co-homology. Categories (syntax) are dual to sets (semantics) -- languages or information. Sheaves are dual to co or dual sheaves. If mathematics is a language then it is dual. "Mathematics is the language of nature" -- Galileo. Mathematics leads to a deep understanding of the physics and hence nature. Structure is dual to function -- protein folding in biology. The shape or structure (syntax) of a protein determines its function or purpose (semantics) -- protein folding is dual. All life is built from proteins hence all life is dual.
Thanos theorem was mind blowing, also this lecture was very important and very well taught. hatts off to you!
Professor eagerly waiting for next video on singular homology
We are currently on winter break. Classes resume next week - and so will the videos!
I am having a doubt at 17:30 For example if I consider simplicial complex spanned by 4 vertices v0,v1,v2,v3, and let vivj denote an oriented edge then the chain v0v1+v1v2+v0v2+2*v2v3-2*v0v3 is a cycle as the boundary is zero but I am not able to see the alpha, beta's etc. in this chain that add to zero that you are talking about at 17:30. Please can you explain. Thanks
Thank you for these excellent lectures!!!. Will the course cover all the four chapters of the Hatcher's book?
We will be away for Christmas holiday, but in January we will resume covering chapter 2. We will likely also cover chapter 3 in the Spring, but may not make it through chapter 4.
happy new year prof. @@MathatAndrews , I m eagerly waiting for the next lectures
Wooho thank you so much please cover the entire book
We plan to cover at least the rest of chapter 2 and then chapter 3.
Thank you so much for the lucid presentation and explanation. Are you going to cover chapter 3 of Hatcher as well ? If you already did, where can I find the video ?
Such a great lecture!!! Merry Christmas🎅🎅🎅
Slowly absorbing the fundamentals of algebraic topology but wondering if it gets expanded from metrics on rings and links on homeological space to all imaginary spaces much as in arithmetic the integers are generalised through fractions to transcendentals and exhaust real number possibilities. Impatient when ploughing through the definitions and their restrictions so wonder if later there is topology using weirder algebras.
Does the definition of a chain imply that the chain can have disconnected components and can have edges with any multiplicity?
The explanation you gave in the beginning about why pi_1 isn't abelian is incorrect and may confuse students. It is clear that a and b^{-1} do not commute because they aren't even loops in the first place, so they don't even represent elements in pi_1. Instead you would have to look at a pair of loops, say ab^{-1} and bc^{-1}, and discuss why they don't commute in pi_1.
hey dear sir as promised i am back to the course .
I will continue algebraic topology after 2 months right now i am off this topic, i stopped at relative homology and exact sequence long and short both
Where did my comment go? I really need an explanation of how these delta complexes are groups. The fundamental group I get; it's the set of loops in a given space that are not equivalent. The Cn and delta n groups seem to be sets of vertices, but what does it _mean?_ Are we still talking about cycles? What is the group's combination operation? What is the inverse of a vertex? I don't get it, please help!
The C_n is the free group generated by the n-simplices. In the case of C_0 its just linear combinations of the vertices. In the case of C_1 its linear combinations of edges. In the case of C_2 its linear combinations of fillled in triangles. The linear combinations themselves don't have to mean anything per se. They're just algebraic objects. Formal sums of n-simplicies but what he's trying to communicate to you in this video is that we can think of linear combinations of these simplicies as "paths" in the complex. In the case of C_1 linear combinations of edges corresponded to paths in the graph. Really the linear combinations don't have to be single paths, they can be disjoint paths. Hence the name chains instead. I'm not sure what you mean by delta_n? Assuming your talking about the boundary operator, its not a group, its a group homomorphism from C_n to C_{n-1}. The way its defined you can see its output is indeed a linear combination of (n-1)-simplicies. Hope this helps!
Topological holes cannot be shrunk down to zero -- non null homotopic.
The big bang is a Janus point/hole (two faces = duality) -- Julian Barbour, physicist.
"Always two there are" -- Yoda.
This what happens when you study too much math
@@mmeister8582 Positive curvature is dual to negative curvature -- Gauss or Riemann curvature.
Contravariant is dual to covariant -- vectors, functors or a dual basis in Riemann geometry.
Riemann geometry contains a hidden dual basis.
Singularities (points) are dual.
Black holes are positive curvature singularities and white holes are negative curvature singularities.
The big bang is an infinite negative curvature singularity -- dark energy is repulsive gravity (inflation).
Gaussian negative curvature is defined using two dual points!
Points are dual to lines -- the principle of duality in geometry.
Homology is dual to co-homology.
Categories (syntax) are dual to sets (semantics) -- languages or information.
Sheaves are dual to co or dual sheaves.
If mathematics is a language then it is dual.
"Mathematics is the language of nature" -- Galileo.
Mathematics leads to a deep understanding of the physics and hence nature.
Structure is dual to function -- protein folding in biology.
The shape or structure (syntax) of a protein determines its function or purpose (semantics) -- protein folding is dual.
All life is built from proteins hence all life is dual.
I am looking from India 🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳🇮🇳
gud morning saaar