This is exactly the kind of content I've been hoping to find as I prepare to start grad school next semester. Your eloquence shows that you clearly know what you are talking about on a deep level much like 3b1b and the like. Thank you for championing intuition focused content. It is often overlooked but it is an important complement to rigorous study. I can't wait for more!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
@@Boarbarktree Convex hulls (converging) are dual to concave hulls (diverging). A convex lens converges or focuses light to a point -- a syntropic process. A concave lens diverges light away from a point -- an entropic process. Generalization is dual to localization. Waves are dual to particles -- quantum duality. Spin statistics theorem:- Waves (Bosons) or symmetric wave functions are dual to particles (Fermions) or anti-symmetric wave functions. Bosons are dual to Fermions. Inside (internal) is dual to outside (external). Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. Potential energy is dual to kinetic energy. Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Space is dual to time -- Einstein. Space is actually dual. Any distance, length is represented by a vector or a convex hull -- non null homotopic. Up is dual to down, left is dual to right, in is dual to out -- space duality.
aaaah the youtube algorithm finally outputs something good. You have my subscription good sir. I had to teach myslef algebraic topology last year for my phd and my tears have still not dried I think. So this is good for me to see again these notions in a big picture
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
I have taken algebraic topology this semester ( my last semester for masters ) and for some reason the professor resigned college and we don't have a professor for AT now. Other students left the course and I decided to study it on my own , these videos are phenomenal . Looking forward to your videos. Thanks in advance.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
I like the visual and auditory style of the video. The handwritten look of it is particularly nice, and I like the combination of scribbles and traditional animation. It's a nice contrast to the dozens of other maths channels which have a much more formal and computerized look, which can sometimes feel alienating. As someone who knows next to nothing about topology, I'm looking forward to your future videos!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Fantastic start! No complaints about the content. As a bit of feedback though I would recommend cutting the first 2:30 right down to sub 30 or sub 10 seconds.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
These animations are lovely and this series is exactly what I was looking for to continue to add to my homological awareness as I have begun reading it in my first year of PhD
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
15:17 So the key is "What is the indexing set A. It is an undefined object. I.e. it is _induced_ by the topology you choose. It could be any set of ordinals, for example. So a toplogy on any set X defines a set A of ordinals? Are they "ordinals induced by X"?
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Exciting! This video gives great intuition and I hope you make more soon; such topics are desperately in need of good motivation and intuition. Maybe tone down the music for future videos though -- I could hear your voice fine but the background music was just a tad overbearing.
I think your visual introduction is terrific! However, at 5 min you have a jump discontinuity in your visual presentation where you introduce the text-based, formal, technical definitions of open sets and continuous functions (which is also beautiful). Just a suggestion, but perhaps, instead of describing these definitions as "unintuitive" and immediately putting off the reader on the technical description, you might say that these definitions can also be introduced in a more visually intuitive and satisfy way (as certainly can be done), which you will do at later time.
Since I made this video I figured out a really good way to derive the axioms for topology and I plan to do a video about it some day. Thanks for your feedback!
Actually homology is one of those subjects that when I first studied I was like: "Well, that's exactly how I would have done it". Then there are always mathematical details to be fixed, but the idea behind mathematics are always intuitive for me; less so is physics.
I think it's because mathematics are in some sense made to feel beautiful, and their formalization is done by men for men. Consequently, mathematics often appear elegant. On the other hand, physics are mostly discovered, which is the reason why they sometimes feel so arbitrary and obscure.
To me the most beautiful thing is the precisely the interplay of mathematics and physics. I like how the sweetest bits of maths have a way of coming around at the most unexpected of places in nature and physics. Like this very topic: I vividly recall the way our former Algebraic Topology prof. turned up all excited one lecture in 2016 just after the Nobel prize had been awarded telling us that she was still abuzz because she never thought she'd live to see the day her studies would apply to "the actual world out there"! :D
I must admit that I am one of those students that finds homology to be rather confusing. I hope this video series will help me to build a better intuition for it.
Similarity, equivalence = duality. Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
This was such an intriguing and well-made video! Can I ask how you are animating? It was such a beautiful style in between hand drawn and clean animated that made it seem like my math notes were coming to life!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
This is so cool. Btw, I'm really courious, what do you use to animate these? Because I can't imagine how do you make these hand-drawn mathematical precise objects
What software do you use for animation? I always want to learn to better explain my ideas, and animations are incredibly powerful for all sorts of mathematical and physics intuition.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
I always wondered if a line R could be deformed in the plane R ^ 2 in such a way that an isomorphism between the line and the curve will be "preserved". Apparently this is topology. Thanks.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
This video is 11:11 long. 1 is a very special number. So as 1111. 1111has only two factors. 11 and 101. This video is also 671 seconds. That factors into 11 and 61.
Great video! I just learned some basic homological algebra this year and I did it in a kind of hard way(in an abelian category) because I don't have any knowledge of algebraic topology. Hopefully, I can pick up some intuitions in the upcoming videos.
@@srijitsanyal3784 I guess it is a combination of course structure at our grad school and covid. It was taught in a very self-contained way so that we don't need algebraic topology and I was blessed with a great lecturer. I don't think I would attempt to do it this way on my own without any guidance. I am not saying it is a bad way I actually enjoyed it but I think some knowledge in algebraic topology would make it easier to understand I guess.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
In the very beginning of this video, it seems you have made an animation based on tiling the plane with square tiles that are homeomorphic to the real projective plane. What did you use to get this animation? I would love to play around with something like that.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Amazing video. How did you do the animations? You say that it is provable that there can't be a classification of all topological spaces, can you cite a reference to that?
The problem is that a classification of all topological spaces would include a classification of any subset of topological spaces - and one type of topological space, namely 4-manifolds, has been shown to be unclassifiable, because a classification would solve a problem in algebra that is known to be unsolvable. For details: core.ac.uk/download/pdf/82780283.pdf
Hmm. Good idea. I don't think I want to fully invert the color scheme but I can dim it down. I'll try that on the next one, let me know how you find it :)
i really like the paper/whiteboard vibes that a white background has, it's such a more intimate and concrete feel compared to the typical black backgrounds
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Dang that ending was like a major twist at the end of a movie. I'm all thinking we are headed towards simplicial homology for video purposes, and then he hits us with singular. I can't wait to see how this develops.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Haha is that your entire map of mathematics at 5:45? So you've got algebra (specifically groups and coordinate rings) on one side, geometry dual to it on the other, analysis to break it all down into points and numbers, and topology as its "synthetic" complement that patches it all together? (also synthetically in the sense of being invariant to charts and triangulations.) Don't know why you've put the whole thing inside of combinatorics though - or did you just not know where to put that?
in all seriousness though, that was never meant to be an accurate map of mathematics. The only part where the positioning was purposeful was algebra and geometry being opposite. And the combinatorics thing was a half joke. Half because lots of intense math can actually be reduced to combinatorics. The cellular approximation theorem, for instance, tells us that algebraic topology can actually be fully reduced to combinatorics
@@Boarbarktree Y = X. Y is equal to X. Y is the same, similar or equivalent to X. Y is dual to X. Same is dual to difference. All mathematical equations are dualities! Homotopic equivalence = duality.
Homology is dual to co-homology. Continuous (topology) is dual to discrete (quantum). Homotopic equivalence = duality. Points are dual to lines -- the principle of duality in geometry. Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality. The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist. Topological holes cannot be shrunk down to zero. Union is dual to intersection. Injective is dual to surjective synthesizes bi-jective or isomorphism. Convex (convergence, syntropy) is dual to concave (divergence, entropy). Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). "You must unlearn all that you have learned" -- Yoda. Learn is dual to unlearn. "Always two there are" -- Yoda. "The Force" = Duality, the Jedi worship duality!
Shout out to my patreoOoOOn *ENDS*
I'll be your patreonian, ... 🧺 🏀... Kia ora, ... 🍊
This is exactly the kind of content I've been hoping to find as I prepare to start grad school next semester. Your eloquence shows that you clearly know what you are talking about on a deep level much like 3b1b and the like. Thank you for championing intuition focused content. It is often overlooked but it is an important complement to rigorous study. I can't wait for more!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
0.o
@@Boarbarktree Convex hulls (converging) are dual to concave hulls (diverging).
A convex lens converges or focuses light to a point -- a syntropic process.
A concave lens diverges light away from a point -- an entropic process.
Generalization is dual to localization.
Waves are dual to particles -- quantum duality.
Spin statistics theorem:- Waves (Bosons) or symmetric wave functions are dual to particles (Fermions) or anti-symmetric wave functions.
Bosons are dual to Fermions.
Inside (internal) is dual to outside (external).
Energy is dual to mass -- Einstein.
Dark energy is dual to dark matter.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
Potential energy is dual to kinetic energy.
Apples fall to the ground because they are conserving duality.
Action is dual to reaction -- Sir Isaac Newton.
Space is dual to time -- Einstein.
Space is actually dual. Any distance, length is represented by a vector or a convex hull -- non null homotopic.
Up is dual to down, left is dual to right, in is dual to out -- space duality.
aaaah the youtube algorithm finally outputs something good. You have my subscription good sir.
I had to teach myslef algebraic topology last year for my phd and my tears have still not dried I think.
So this is good for me to see again these notions in a big picture
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
._.
Wow. This is amazing, I can't wait for more.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Actually laughed out loud when I saw combinatorics containing everything else. What a great subtle joke.
He did also joke about algebra and geometry
I dont get it
Love the "shoutout to my patreon(s??)" at the end hahaha.
Great vid bud
I have taken algebraic topology this semester ( my last semester for masters ) and for some reason the professor resigned college and we don't have a professor for AT now.
Other students left the course and I decided to study it on my own , these videos are phenomenal . Looking forward to your videos.
Thanks in advance.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Can't help squeeing when I saw commutative diagram, this gonna be good!
Came here from Grant Sanderson
I like the visual and auditory style of the video. The handwritten look of it is particularly nice, and I like the combination of scribbles and traditional animation. It's a nice contrast to the dozens of other maths channels which have a much more formal and computerized look, which can sometimes feel alienating.
As someone who knows next to nothing about topology, I'm looking forward to your future videos!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
quick question: at 6:40 you say you can prove it is impossible to classify all topological spaces - could you expand on that slightly?
Fantastic start! No complaints about the content. As a bit of feedback though I would recommend cutting the first 2:30 right down to sub 30 or sub 10 seconds.
my one regret
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
omg who are you @Blaze Thrower. Nice video btw @Boarbarktree ~
@@Leo-pm1kp Get a better profile picture ;)
Excited for what’s coming
Absolutely incredible - can’t wait to follow along!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Great introduction, I'm looking forward to the rest of this series
Fantastic approach to a wonderful topic with not enough accessible educational materials available for it. Hope this channel thrives!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
These animations are lovely and this series is exactly what I was looking for to continue to add to my homological awareness as I have begun reading it in my first year of PhD
occasionally you watch a video and just have to instantly sub. that's this one.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
This looks nuts, keep it up!! Can't wait
What do you use for the animations? Looks pretty amazing.
Category theory creeping in!
15:17 So the key is "What is the indexing set A. It is an undefined object. I.e. it is _induced_ by the topology you choose. It could be any set of ordinals, for example. So a toplogy on any set X defines a set A of ordinals? Are they "ordinals induced by X"?
Can you then ask: "So what is the set X which is closed when you use the topology induced by X on itself?" Or is that in some sense transcendental?
Wow, your animation style is really nice! I like the way you animated the mappings using drawings - did you draw every frame?
For most of the actual animations, I roughly draw every frame and then edit the strokes to fine tune the motion
Currently taking algebraic Topology so I am seriously looking forward to this!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Awesome to see you making content my bro! Looking forward to seeing more in the future 🙂
Exciting! This video gives great intuition and I hope you make more soon; such topics are desperately in need of good motivation and intuition. Maybe tone down the music for future videos though -- I could hear your voice fine but the background music was just a tad overbearing.
thank you for the constructive feedback 😁
Man thanks for all your efforts! i hope you will explain general topology or a topology too
thanks for the shout out
I think your visual introduction is terrific! However, at 5 min you have a jump discontinuity in your visual presentation where you introduce the text-based, formal, technical definitions of open sets and continuous functions (which is also beautiful). Just a suggestion, but perhaps, instead of describing these definitions as "unintuitive" and immediately putting off the reader on the technical description, you might say that these definitions can also be introduced in a more visually intuitive and satisfy way (as certainly can be done), which you will do at later time.
Since I made this video I figured out a really good way to derive the axioms for topology and I plan to do a video about it some day. Thanks for your feedback!
Actually homology is one of those subjects that when I first studied I was like: "Well, that's exactly how I would have done it".
Then there are always mathematical details to be fixed, but the idea behind mathematics are always intuitive for me; less so is physics.
I think it's because mathematics are in some sense made to feel beautiful, and their formalization is done by men for men. Consequently, mathematics often appear elegant. On the other hand, physics are mostly discovered, which is the reason why they sometimes feel so arbitrary and obscure.
To me the most beautiful thing is the precisely the interplay of mathematics and physics. I like how the sweetest bits of maths have a way of coming around at the most unexpected of places in nature and physics. Like this very topic: I vividly recall the way our former Algebraic Topology prof. turned up all excited one lecture in 2016 just after the Nobel prize had been awarded telling us that she was still abuzz because she never thought she'd live to see the day her studies would apply to "the actual world out there"! :D
@Kmeleon very ironic that you would say that while the group theoretic nature of homology was discovered by a woman
@@Boarbarktree Sorry, when I said men, I meant humans. English is not my first language.
Don't worry about it, I just never pass on an opportunity to give credit to the incredible Emmy Noether 😊
there is a mistake in definition 1.7.1 it is f^(-1)(U) not f^(-1)(X)
5.26
I must admit that I am one of those students that finds homology to be rather confusing. I hope this video series will help me to build a better intuition for it.
Similarity, equivalence = duality.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
This was such an intriguing and well-made video! Can I ask how you are animating? It was such a beautiful style in between hand drawn and clean animated that made it seem like my math notes were coming to life!
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
5:02 , Definition 1.7.1. is broken, should say that f^-1(U) is open in X, not that f^-1(X) is open in X.
Wow! Good spotting! This is from some notes I wrote several years ago and I never noticed the typo!
What a great channel name, Bourbaki
What's up next?
Complex group theory!
Sad this channel never took off!
Amazing. Thank you
This is so cool. Btw, I'm really courious, what do you use to animate these? Because I can't imagine how do you make these hand-drawn mathematical precise objects
Any idea anyone?
Keep up with this. Amazing video.
I haven't seen the video yet, subscribed for the channel name :)
Thanks for the great video
Good video. Keep it up . Make more such videos
Absolutely FANTABULOUSSSSSSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Dear sir, you have earned a sub, also how do you make these videos?!!! :)
Definition 1.7.1 at 5:22 should be "...the inverse image f^{-1}(U) is an open set in X..."
Good eye 😉
Man this is beautiful!! Keep it up please! Lov this for real
What software do you use for animation? I always want to learn to better explain my ideas, and animations are incredibly powerful for all sorts of mathematical and physics intuition.
agreed, im super curious as well
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Music is a bit too loud at some points but video is very clear and interesting, thanks for this series!!
Yup... From part 2 both the background music and brightness are turned down so it should be easier on the senses
I'm fully in love with your content!
11:02 commutative diagrams 😳
I love this! I'll do anything to learn all this!
This is the next big thing, I called it first, remember me later
Become a patron then ;)
@@Boarbarktree Never done that. Guess I'll have to... :-)
@@Boarbarktree Oh geez, can I not just send money without paying monthly? :O
@@nexovec sure you can! ko-fi.com/boarbarktree
What's with the two notes of Chopin at the start?
it's so everyone in the room turns and looks because they know something amazing is about to happen
Loved this video, thank you!
any link to the background music?
Cool! Waiting for more :)
I always wondered if a line R could be deformed in the plane R ^ 2 in such a way that an isomorphism between the line and the curve will be "preserved". Apparently this is topology. Thanks.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Love the squiggles
Does Taurus mean Doughnut in NedKellyian
5:40 we know exactly what's at the top!
ABSOLUTELY BEAUTIFUL!
This video is 11:11 long. 1 is a very special number. So as 1111. 1111has only two factors. 11 and 101. This video is also 671 seconds. That factors into 11 and 61.
did you make the background music?
Nah pinched it off the free youtube music library 😅
@@Boarbarktree Nice lo
You had me at "Boarbarktree"!
Excellent video!
thank you Nicolas, very cool
Omg sjdjwokcfjure I can't get over how beautiful the perspective of 3:08 about continuity is
Great video! I just learned some basic homological algebra this year and I did it in a kind of hard way(in an abelian category) because I don't have any knowledge of algebraic topology. Hopefully, I can pick up some intuitions in the upcoming videos.
but why would you do that
to yourself
@@srijitsanyal3784 I guess it is a combination of course structure at our grad school and covid. It was taught in a very self-contained way so that we don't need algebraic topology and I was blessed with a great lecturer. I don't think I would attempt to do it this way on my own without any guidance. I am not saying it is a bad way I actually enjoyed it but I think some knowledge in algebraic topology would make it easier to understand I guess.
@@srijitsanyal3784 it happens to the best of us
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
As you write the diagram in 11:00, i had flashbacks from higher category theory.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
I need more videos.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
In the very beginning of this video, it seems you have made an animation based on tiling the plane with square tiles that are homeomorphic to the real projective plane. What did you use to get this animation? I would love to play around with something like that.
I drew it lol, in Adobe animate
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Amazing video. How did you do the animations?
You say that it is provable that there can't be a classification of all topological spaces, can you cite a reference to that?
The problem is that a classification of all topological spaces would include a classification of any subset of topological spaces - and one type of topological space, namely 4-manifolds, has been shown to be unclassifiable, because a classification would solve a problem in algebra that is known to be unsolvable. For details: core.ac.uk/download/pdf/82780283.pdf
@@Boarbarktree thank you very much
Wouldn't be a set of all sets kind of thing?
What do you use for animations? Thank you
are these animations hand drawn?
Amazing work
Great video!
Thank you for your work
this is so cool
Beautiful! Subscribed.
could you use a black background ? it's easier on the eyes
I prefer black background with white ink as well. Most of the big youtube maths channels use dark themes or cardboard.
Hmm. Good idea. I don't think I want to fully invert the color scheme but I can dim it down. I'll try that on the next one, let me know how you find it :)
i really like the paper/whiteboard vibes that a white background has, it's such a more intimate and concrete feel compared to the typical black backgrounds
what animation software are you using? ive been thinking about starting a channel like this too!
"natural choices" kek
Thanks KneeCoalLass BoarBarkTree
🤣
Woah, I see why people love math now. This just made a bunch of things click. thank you.
I will invent something, though.
Woah, supernice vid
My topology Professor mentioned homology and here I am after a few TH-cam searches
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Shit. I just learned what Celtic Gold really is!❤😂
How do you know I didn't?
Because Henri Poincare was not Swedish. (not even mod null sets)
@@srijitsanyal3784 swedish is actually a strictly stronger condition than almost swedish. Polish spaces, for instance, are counter-examples.
that joke would be a lot funnier if poland was actually anywhere near sweden
Cool!
P-ha-hah, combinatorics encompasses all of math, nice touch
Hello, Bourbaki's cousin Boarbarktree!
Like the channel, and the name is funny xD
You Could Have Invented Homology ... but didn't
Dang that ending was like a major twist at the end of a movie. I'm all thinking we are headed towards simplicial homology for video purposes, and then he hits us with singular. I can't wait to see how this develops.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!
Here in 2020 before this guy blows up like 3b1b, mark my fucking words
Seems like a lot of obsession over holes though.
from 3blue1brown
Yup. It is gonna blow up!
Haha is that your entire map of mathematics at 5:45? So you've got algebra (specifically groups and coordinate rings) on one side, geometry dual to it on the other, analysis to break it all down into points and numbers, and topology as its "synthetic" complement that patches it all together? (also synthetically in the sense of being invariant to charts and triangulations.) Don't know why you've put the whole thing inside of combinatorics though - or did you just not know where to put that?
All math is combinatorics actually
@@Boarbarktree math is the art of not having to do combinatorics
in all seriousness though, that was never meant to be an accurate map of mathematics. The only part where the positioning was purposeful was algebra and geometry being opposite. And the combinatorics thing was a half joke. Half because lots of intense math can actually be reduced to combinatorics. The cellular approximation theorem, for instance, tells us that algebraic topology can actually be fully reduced to combinatorics
@@Boarbarktree Y = X.
Y is equal to X. Y is the same, similar or equivalent to X. Y is dual to X.
Same is dual to difference.
All mathematical equations are dualities!
Homotopic equivalence = duality.
@@srijitsanyal3784 OK but then combinatorics is the art of not having to do combinatorics.
I _did_ invent homology ... or something like it.
Hehehe @ algebra/geometry
Starts at 2:38. Life is too short for verbose intros.
Homology is dual to co-homology.
Continuous (topology) is dual to discrete (quantum).
Homotopic equivalence = duality.
Points are dual to lines -- the principle of duality in geometry.
Null homotopic implies contraction to a point where as non null homotopic requires at least two points (a line) or duality.
The big bang is a Janus hole/point (two faces = duality) -- Julian Barbour, physicist.
Topological holes cannot be shrunk down to zero.
Union is dual to intersection.
Injective is dual to surjective synthesizes bi-jective or isomorphism.
Convex (convergence, syntropy) is dual to concave (divergence, entropy).
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
"You must unlearn all that you have learned" -- Yoda.
Learn is dual to unlearn.
"Always two there are" -- Yoda.
"The Force" = Duality, the Jedi worship duality!