To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/Mathemaniac/. You’ll also get 20% off an annual premium subscription. Next video should be about a genius proof of hairy ball theorem (in fact, proof of the more general Poincaré-Hopf theorem). That will be out next year (I haven't made the video yet)!
How the heck you can produce content this quickly, of this quality, is remarkable. Loved your Euler char video regarding its range (rather than the strict equality). Keep it up!
The last point (generalisation to higher dim) is extraordinarily important in theoretical physics. The Euler characteristic of the manifold is closely contingent to the conformal anomaly when coupling a conformal field theory to this certain background. The anomalous characteristic monotonically decreases as the energy scale goes down. Such statement is proved in 2D(1989) and 4D(2011) which reflects the thermodynamic property of quantum field theory. However, the statement cannot be established in 3D precisely because of the vanishing Euler characteristic in odd dimension, and a lot of our condensed matter models are in 3D. If mathematicians find a way to generate non trivial topological characteristics in 3D, it might open the door to resolving such conundrum . We live in an amazing era of rapidly advancing mathematics and theoretical physics, with so much more yet to be discovered!
Actually if you asked me my favourite theorem of all time a year ago, I would have put Gauss-Bonnet as a close second (Theorema Egregium is the first), but because of the relation to physics, this is now my favourite theorem of all time!
Theorem: The Euler characteristic of a topological space is a topological invariant. Proof: Let X be a topological space, and let χ(X) be its Euler characteristic, defined as: χ(X) = Σ_i (-1)^i β_i where β_i is the i-th Betti number of X, which counts the number of i-dimensional "holes" in X. To prove that χ(X) is a topological invariant, we need to show that it remains unchanged under continuous deformations of X, such as stretching, twisting, or bending, but not tearing or gluing. Consider a continuous map f : X → Y between two topological spaces X and Y. The induced homomorphisms on the homology groups of X and Y satisfy the following property: f_* : H_i(X) → H_i(Y) is a group homomorphism for each i Moreover, the alternating sum of the ranks of these homomorphisms is equal to the Euler characteristic: Σ_i (-1)^i rank(f_*) = χ(X) - χ(Y) Now, if f is a homeomorphism, i.e., a continuous bijection with a continuous inverse, then the induced homomorphisms f_* are isomorphisms, and their ranks are equal to the Betti numbers of X and Y: rank(f_*) = β_i(X) = β_i(Y) for each i Therefore, we have: Σ_i (-1)^i rank(f_*) = Σ_i (-1)^i β_i(X) - Σ_i (-1)^i β_i(Y) = χ(X) - χ(Y) = 0 This implies that χ(X) = χ(Y) whenever X and Y are homeomorphic, i.e., χ is a topological invariant. This proof highlights the fundamental role of the Euler characteristic in capturing the essential topological properties of a space, and suggests that the concept of zero or nothingness may be intimately connected to the deep structure of space and time.
This video was beautiful, especially when it all comes together at the end. I literally gasped at 18:41 and just had to write a comment. Your animations are honestly extraordinary and I love your videos.
Beautiful, as someone who has not yet studied Differential Geometry I could still follow along the intuition. Really loved how it all came together at the end with the euler characteristic bit, it was like a well written movie script!!❤
Amazing video as always! Differential geometry is the perfect topic to do videos like this for so I don't know why we don't get more people doing so. You are definitely my favourite maths UA-camr.
Wonderful video as always.. lol . And as usual, ill probably need to watch it 3 times cause large chunks go over my dense head unless i pause repeatedly
Topological Invariants and the Shape of Space: Topology is the branch of mathematics that deals with the properties of spaces that are preserved under continuous deformations, such as stretching or twisting, but not tearing or gluing. Topological invariants are quantities that capture the essential features of a space, such as its connectivity, holes, or boundaries, without referring to its specific geometry or dimensions. One of the most fascinating aspects of topological invariants is that they can be used to classify and distinguish spaces that may look very different from each other, but are actually equivalent from a topological perspective. For example, a coffee mug and a donut are topologically equivalent, because they both have a single hole and can be continuously deformed into each other without tearing or gluing. This idea suggests that the fundamental shape of space may not be determined by its specific geometry or dimensions, but rather by its topological properties. In other words, the essential features of space may be captured by its connectivity and holes, rather than its size or curvature. This insight has profound implications for our understanding of the nature of space and time, and may even hint at the possibility of exotic spacetime structures, such as wormholes or higher-dimensional manifolds. By focusing on topological invariants and the intrinsic shape of space, we may be able to develop a more abstract and general understanding of the nature of reality, one that transcends the limitations of specific geometries or dimensions. This approach could lead to new insights into the origin and structure of the universe, and may even suggest novel ways of unifying the fundamental forces of nature.
I don't understand one thing - if we add some nipple to a torus on the side, the surface area certainly will grow, so why doesn't it affect the formula? Do we speak about some kind of regularity or maybe local euclideanness to Gauss-Bonnet to work properly? Like stellarator form is fine, but the cup isn't though both are just weird torii in topological terms? What are the prequisites for a surface?
Surface area grows, but when you stick something to the torus, then the site at which you stick to the torus will have negative curvature. "Surface" already means locally Euclidean, and we do require differentiable and orientable surfaces in order for curvature to even be defined.
@mathemaniac nipple on the side doesn't prevent diffs or normals, it will just "spike" curvature at certain regions. But locally euclidean wasn't obvious - methinks that was never mentioned in previous videos, but I might be wrong. Also a followup - how do we distinguish valid locally eucledean from invalid ones? Will it just loop to "must have area according to gauss-bonne".
Yes - we can guarantee that usually by Picard-Lindelöf theorem or some variants of it. Basically existence of geodesics in a small neighbourhood is asking whether solutions to an initial value problem of some differential equations exist, which can be dealt with by some existence theorems. I don't know exactly the details though.
See Cohn-Vossen's inequality for the non-compact case - it's an inequality instead. However, if you ask me exactly where the argument breaks down, I might point to the subdivision part as well - just think of R^2 itself. Due to the unboundedness, there is no way to properly sum up the contributions of geodesic triangles.
It's been 50 years since I aced the Math SAT, though I could still if I had to duck chalk thrown at me by a Jesuit. Though I fought to get some sense of the magic, this was nearly incomprehensible, and I had to settle for admiring the brilliance of the production. When I think Euler it's Emanuel Handmann's fantastic 1753 portrait that comes to mind. Gauss exists in my head via anecdotes I have read of him as an especially brilliant child. I am left to wonder about the utility of this vein of mathematics, is it for making money, or designing weapons? Seems that those two areas of endeavor eat up all the engineering talent.
![The Dark Matter of AI [Mechanistic Interpretability]](http://i.ytimg.com/vi/UGO_Ehywuxc/mqdefault.jpg)
To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/Mathemaniac/. You’ll also get 20% off an annual premium subscription.
Next video should be about a genius proof of hairy ball theorem (in fact, proof of the more general Poincaré-Hopf theorem). That will be out next year (I haven't made the video yet)!
How the heck you can produce content this quickly, of this quality, is remarkable. Loved your Euler char video regarding its range (rather than the strict equality). Keep it up!
The last point (generalisation to higher dim) is extraordinarily important in theoretical physics. The Euler characteristic of the manifold is closely contingent to the conformal anomaly when coupling a conformal field theory to this certain background. The anomalous characteristic monotonically decreases as the energy scale goes down. Such statement is proved in 2D(1989) and 4D(2011) which reflects the thermodynamic property of quantum field theory. However, the statement cannot be established in 3D precisely because of the vanishing Euler characteristic in odd dimension, and a lot of our condensed matter models are in 3D. If mathematicians find a way to generate non trivial topological characteristics in 3D, it might open the door to resolving such conundrum . We live in an amazing era of rapidly advancing mathematics and theoretical physics, with so much more yet to be discovered!
Actually if you asked me my favourite theorem of all time a year ago, I would have put Gauss-Bonnet as a close second (Theorema Egregium is the first), but because of the relation to physics, this is now my favourite theorem of all time!
Berry Curvature won the Nobel Prize :)
Theorem: The Euler characteristic of a topological space is a topological invariant.
Proof:
Let X be a topological space, and let χ(X) be its Euler characteristic, defined as:
χ(X) = Σ_i (-1)^i β_i
where β_i is the i-th Betti number of X, which counts the number of i-dimensional "holes" in X.
To prove that χ(X) is a topological invariant, we need to show that it remains unchanged under continuous deformations of X, such as stretching, twisting, or bending, but not tearing or gluing.
Consider a continuous map f : X → Y between two topological spaces X and Y.
The induced homomorphisms on the homology groups of X and Y satisfy the following property:
f_* : H_i(X) → H_i(Y) is a group homomorphism for each i
Moreover, the alternating sum of the ranks of these homomorphisms is equal to the Euler characteristic:
Σ_i (-1)^i rank(f_*) = χ(X) - χ(Y)
Now, if f is a homeomorphism, i.e., a continuous bijection with a continuous inverse, then the induced homomorphisms f_* are isomorphisms, and their ranks are equal to the Betti numbers of X and Y:
rank(f_*) = β_i(X) = β_i(Y) for each i
Therefore, we have:
Σ_i (-1)^i rank(f_*) = Σ_i (-1)^i β_i(X) - Σ_i (-1)^i β_i(Y) = χ(X) - χ(Y) = 0
This implies that χ(X) = χ(Y) whenever X and Y are homeomorphic, i.e., χ is a topological invariant.
This proof highlights the fundamental role of the Euler characteristic in capturing the essential topological properties of a space, and suggests that the concept of zero or nothingness may be intimately connected to the deep structure of space and time.
This video was beautiful, especially when it all comes together at the end. I literally gasped at 18:41 and just had to write a comment. Your animations are honestly extraordinary and I love your videos.
Thank you so much!
Best Christmas present.
Glad you like it!
One of my favorite theorems. Thanks for the Christmas gift!
Glad you like it!
This was a super satisfying follow-up to both the video on Theorema Egregium and the one on Euler characteristic!
Beautiful, as someone who has not yet studied Differential Geometry I could still follow along the intuition. Really loved how it all came together at the end with the euler characteristic bit, it was like a well written movie script!!❤
Amazing video as always! Differential geometry is the perfect topic to do videos like this for so I don't know why we don't get more people doing so. You are definitely my favourite maths UA-camr.
Thank you so much!
A wonderful holiday gift, bravo @Mathemaniac
Hope you like it :)
A really good set of explanations, thank you and well done!
Thanks!
Very good description!
super underrated channel!
Thanks!
Wonderful video as always.. lol . And as usual, ill probably need to watch it 3 times cause large chunks go over my dense head unless i pause repeatedly
Topological Invariants and the Shape of Space:
Topology is the branch of mathematics that deals with the properties of spaces that are preserved under continuous deformations, such as stretching or twisting, but not tearing or gluing. Topological invariants are quantities that capture the essential features of a space, such as its connectivity, holes, or boundaries, without referring to its specific geometry or dimensions.
One of the most fascinating aspects of topological invariants is that they can be used to classify and distinguish spaces that may look very different from each other, but are actually equivalent from a topological perspective. For example, a coffee mug and a donut are topologically equivalent, because they both have a single hole and can be continuously deformed into each other without tearing or gluing.
This idea suggests that the fundamental shape of space may not be determined by its specific geometry or dimensions, but rather by its topological properties. In other words, the essential features of space may be captured by its connectivity and holes, rather than its size or curvature. This insight has profound implications for our understanding of the nature of space and time, and may even hint at the possibility of exotic spacetime structures, such as wormholes or higher-dimensional manifolds.
By focusing on topological invariants and the intrinsic shape of space, we may be able to develop a more abstract and general understanding of the nature of reality, one that transcends the limitations of specific geometries or dimensions. This approach could lead to new insights into the origin and structure of the universe, and may even suggest novel ways of unifying the fundamental forces of nature.
great description...thanks
Very nice! Thank you!
I don't understand one thing - if we add some nipple to a torus on the side, the surface area certainly will grow, so why doesn't it affect the formula? Do we speak about some kind of regularity or maybe local euclideanness to Gauss-Bonnet to work properly? Like stellarator form is fine, but the cup isn't though both are just weird torii in topological terms? What are the prequisites for a surface?
Surface area grows, but when you stick something to the torus, then the site at which you stick to the torus will have negative curvature. "Surface" already means locally Euclidean, and we do require differentiable and orientable surfaces in order for curvature to even be defined.
@mathemaniac nipple on the side doesn't prevent diffs or normals, it will just "spike" curvature at certain regions. But locally euclidean wasn't obvious - methinks that was never mentioned in previous videos, but I might be wrong. Also a followup - how do we distinguish valid locally eucledean from invalid ones? Will it just loop to "must have area according to gauss-bonne".
@DeathSugar Well, tbf it's supposed to be a manifold, as stated by the M in the integral
Differential geometry is badass
Nice. Hidden in here are paths to Lie algebra, and why covering a simple closed surface with hexagons will require pentagons, too.
This basically is the reason you can eat pizza holding the crust with merely a finger. Good video!
You are confusing with the other theorem called Theorema Egregium, which is the theorem that is often introduced with pizza-eating.
🤣🤣
It has been a little while since I have read a book by Presley. Fun read while I got the chance for an independent study.
Is any deformed manifold could be triangulated by finitely many geodesic triangles ?
Yes - we can guarantee that usually by Picard-Lindelöf theorem or some variants of it. Basically existence of geodesics in a small neighbourhood is asking whether solutions to an initial value problem of some differential equations exist, which can be dealt with by some existence theorems. I don't know exactly the details though.
What breaks without compactness, the subdivision?
See Cohn-Vossen's inequality for the non-compact case - it's an inequality instead. However, if you ask me exactly where the argument breaks down, I might point to the subdivision part as well - just think of R^2 itself. Due to the unboundedness, there is no way to properly sum up the contributions of geodesic triangles.
@mathemaniac Very interesting. Somehow you can hide away positive curvature "at infinity" but you can't do the same with negative curvature.
how about a proof that we can't split the sphere into less than 8 triangles. Or can we ?
I don't know of such a theorem.
Ok, now do the generalized version.
Edit: oh you discussed it
Is it negative though
It looks like an interlude and the riddler
Wow~!
Is this a version of king of the hill
It's been 50 years since I aced the Math SAT, though I could still if I had to duck chalk thrown at me by a Jesuit. Though I fought to get some sense of the magic, this was nearly incomprehensible, and I had to settle for admiring the brilliance of the production.
When I think Euler it's Emanuel Handmann's fantastic 1753 portrait that comes to mind. Gauss exists in my head via anecdotes I have read of him as an especially brilliant child. I am left to wonder about the utility of this vein of mathematics, is it for making money, or designing weapons? Seems that those two areas of endeavor eat up all the engineering talent.
I can watch this in German!!!
Yeah recently UA-cam autodubs, but please let me know if the autodubbing is at least passable.
What is delta