6 ms·

Mathematicians Identify Threshold at Which Shapes Give Way

100 points by nsoonhui 5y ago

Ericson2314 5y ago
Quanta Magazine nails it yet again. The connection to turbulence is also very interesting. Only question is what the C ^ x,y notation is about, as they only explained one axis. Anyone know?
- Ericson2314 5y ago
  Also hah! I saw someone once counter, "If category theory is so universally useful, tell me how it helps the study of differential equations?" The ISA "sprint" https://www.ias.edu/math/sp/hprinciple_flex_geometry_pdes https://www.ias.edu/math/sp/hprinciple_flex_geometry_pdes referenced by the article mentions PDEs and Fukaya categories, which I hadn't heard of, so hah! Just don't ask me to explain it :D
- rrmm 5y ago
  Holder spaces. https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pdf https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pd... """ Let Ω be an open set in Rn, 0<α≤1, and k a nonnegative integer. The (uniform) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are uniformly Holder continuous with exponent α in Ω. The(local) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are locally Holder continuous with exponent α """ The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. Also: https://en.wikipedia.org/wiki/H%C3%B6lder_condition https://en.wikipedia.org/wiki/H%C3%B6lder_condition
  - vitus 5y ago
    > The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous." Normally, you'd have something like For all ϵ > 0, there exists δ(x, ϵ) > 0 such that if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ. (namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.) But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away. For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x) In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.
    - jbrot 5y ago
      Small correction: the definition you shared is that of “uniform continuity” [1] not “absolute continuity” [2]. Absolute continuity is a slightly stronger condition where instead of providing a single (x1, x2) pair with |x1 - x2| < delta, you can now provide a finite list of pairs. [1] https://en.m.wikipedia.org/wiki/Uniform_continuity https://en.m.wikipedia.org/wiki/Uniform_continuity [2] https://en.m.wikipedia.org/wiki/Absolute_continuity https://en.m.wikipedia.org/wiki/Absolute_continuity
      - vitus 5y ago
        Ah, good catch! The wiki quote that I cited does reference uniform continuity, so the rest still holds. (left for posterity so your comment makes sense)
  - Ericson2314 5y ago
    Thank you!! I guess in hindsight they did talk about the exponents and the iterated differentiabilty, but I didn't make the connection. I think this was at the end of the analysis textbook too, in one of those "looking ahead future topics" sections.
adamnemecek 5y ago
Embeddings were not invented by Nash, were they?
- kkylin 5y ago
  No, but Nash was the first (AFAIK) to prove that every Riemannian manifold can be isometrically embedded in an Euclidean space of sufficiently high dimension. See, e.g., [1,2]. Previous embedding theorems (see Whitney embedding [3]) preserved topology but not necessarily geometry. [1] https://en.wikipedia.org/wiki/Embedding#Riemannian_and_pseudo-Riemannian_geometry https://en.wikipedia.org/wiki/Embedding#Riemannian_and_pseud... [2] https://en.wikipedia.org/wiki/Nash_embedding_theorem https://en.wikipedia.org/wiki/Nash_embedding_theorem [3] https://en.wikipedia.org/wiki/Whitney_embedding_theorem https://en.wikipedia.org/wiki/Whitney_embedding_theorem
  - rsj_hn 5y ago
    > Previous embedding theorems (see Whitney embedding [3]) preserved topology but not necessarily geometry. The definition of Riemmannian manifold fairly straightforwardly means that every compact smooth (I think you only need C^2) manifold can be isometrically embedded in a high enough dimensional euclidean space (with the euclidean metric) just by using the fact that every riemannian manifold is locally euclidean, which means we can cover it with balls on which there are local isomorphisms, and from these we can pick a finite subcover and partition of unity that glues together for a global isometry. Basically you are gluing together local solutions to a PDE. This is something often covered in a first year course on geometry. For the details of this approach worked out, see this expository paper: https://www.math.mcgill.ca/gantumur/math580f12/siyuan.lu.pdf https://www.math.mcgill.ca/gantumur/math580f12/siyuan.lu.pdf To the best of my knowledge, this result doesn't even have a name attached to it (I could be wrong, but it was always presented as just a series of unnamed propositions whenever I saw this). That's not the hard part, the hard part is putting an explicit bound on the size of the target space other than "finite". So the genius of Nash was to put a strong bound on the dimension of the target space as well as the size of the target space. Rather than using general compactness and existence arguments, this requires careful arguments and innovative geometric ideas and is really a suprising result.
sean_pedersen 5y ago
0.0?
ubavic 5y ago
There are some misleading sentences in this article. > The C stands for continuity and the superscript zero means the curves on the embedded surface have no derivatives, not even a first. This is not exactly true. C maps maybe have higher derivatives and maybe they don’t. Certainly, C isn’t a class of only continuous mappings that don’t have the first derivative. Also, the example with the letter U is bad (it tries to make a point, and it fails in doing so). A U-shaped curve can be perfectly smooth at all points. It would be “problematic” if we, for example, made the letter U by “glueing” a part of the circle and two segments.
SOFAYON 5y ago
>A key limitation of Cao and Inauen’s paper is that it requires embedding a shape into eight-dimensional space, rather than the three-dimensional space Gromov had in mind. With extra dimensions, the mathematicians gained more room to add twists, which made the problem easier. Mathematicians Identify Threshold at Which Shapes Give Way [In 8 Dimensions!]

100 points by nsoonhui 5y ago

Quanta Magazine nails it yet again. The connection to turbulence is also very interesting. Only question is what the C ^ x,y notation is about, as they only explained one axis. Anyone know?

Ericson2314 5y ago
Also hah! I saw someone once counter, "If category theory is so universally useful, tell me how it helps the study of differential equations?" The ISA "sprint" https://www.ias.edu/math/sp/hprinciple_flex_geometry_pdes https://www.ias.edu/math/sp/hprinciple_flex_geometry_pdes referenced by the article mentions PDEs and Fukaya categories, which I hadn't heard of, so hah! Just don't ask me to explain it :D
rrmm 5y ago
Holder spaces. https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pdf https://math.ucdenver.edu/~jmandel/classes/7760f05/spaces.pd... """ Let Ω be an open set in Rn, 0<α≤1, and k a nonnegative integer. The (uniform) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are uniformly Holder continuous with exponent α in Ω. The(local) Holder spaces Ck,α(Ω) consist of functions whose k−th order derivatives are locally Holder continuous with exponent α """ The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. Also: https://en.wikipedia.org/wiki/H%C3%B6lder_condition https://en.wikipedia.org/wiki/H%C3%B6lder_condition
- vitus 5y ago
  > The Holder continuity condition looks sorta like the limit definition of a derivative with an exponent on the denominator. This is actually stronger than the epsilon-delta definition of continuity, as noted in the wiki article -- "For any α > 0, the condition implies the function is uniformly continuous." Normally, you'd have something like For all ϵ > 0, there exists δ(x, ϵ) > 0 such that if |x - x₀| < δ, then |f(x) - f(x₀)| < ϵ. (namely, at any given point, there's a ball of nearby points such that all their corresponding outputs are close together.) But, with absolute continuity, δ is purely a function of ϵ, namely, that ball doesn't change size even as we move further away. For instance, f(x)=x is absolutely continuous, while f(x)=x^2 is not. (for the former case, we can use δ = ϵ/2; for the latter case, we'd need a δ that looks something like 2ϵ·x) In this case, since the Holder condition depends only on the distance between x and y, it automatically implies absolute continuity.
  - jbrot 5y ago
    Small correction: the definition you shared is that of “uniform continuity” [1] not “absolute continuity” [2]. Absolute continuity is a slightly stronger condition where instead of providing a single (x1, x2) pair with |x1 - x2| < delta, you can now provide a finite list of pairs. [1] https://en.m.wikipedia.org/wiki/Uniform_continuity https://en.m.wikipedia.org/wiki/Uniform_continuity [2] https://en.m.wikipedia.org/wiki/Absolute_continuity https://en.m.wikipedia.org/wiki/Absolute_continuity
    - vitus 5y ago
      Ah, good catch! The wiki quote that I cited does reference uniform continuity, so the rest still holds. (left for posterity so your comment makes sense)
- Ericson2314 5y ago
  Thank you!! I guess in hindsight they did talk about the exponents and the iterated differentiabilty, but I didn't make the connection. I think this was at the end of the analysis textbook too, in one of those "looking ahead future topics" sections.

adamnemecek 5y ago

Embeddings were not invented by Nash, were they?

kkylin 5y ago
No, but Nash was the first (AFAIK) to prove that every Riemannian manifold can be isometrically embedded in an Euclidean space of sufficiently high dimension. See, e.g., [1,2]. Previous embedding theorems (see Whitney embedding [3]) preserved topology but not necessarily geometry. [1] https://en.wikipedia.org/wiki/Embedding#Riemannian_and_pseudo-Riemannian_geometry https://en.wikipedia.org/wiki/Embedding#Riemannian_and_pseud... [2] https://en.wikipedia.org/wiki/Nash_embedding_theorem https://en.wikipedia.org/wiki/Nash_embedding_theorem [3] https://en.wikipedia.org/wiki/Whitney_embedding_theorem https://en.wikipedia.org/wiki/Whitney_embedding_theorem
- rsj_hn 5y ago
  > Previous embedding theorems (see Whitney embedding [3]) preserved topology but not necessarily geometry. The definition of Riemmannian manifold fairly straightforwardly means that every compact smooth (I think you only need C^2) manifold can be isometrically embedded in a high enough dimensional euclidean space (with the euclidean metric) just by using the fact that every riemannian manifold is locally euclidean, which means we can cover it with balls on which there are local isomorphisms, and from these we can pick a finite subcover and partition of unity that glues together for a global isometry. Basically you are gluing together local solutions to a PDE. This is something often covered in a first year course on geometry. For the details of this approach worked out, see this expository paper: https://www.math.mcgill.ca/gantumur/math580f12/siyuan.lu.pdf https://www.math.mcgill.ca/gantumur/math580f12/siyuan.lu.pdf To the best of my knowledge, this result doesn't even have a name attached to it (I could be wrong, but it was always presented as just a series of unnamed propositions whenever I saw this). That's not the hard part, the hard part is putting an explicit bound on the size of the target space other than "finite". So the genius of Nash was to put a strong bound on the dimension of the target space as well as the size of the target space. Rather than using general compactness and existence arguments, this requires careful arguments and innovative geometric ideas and is really a suprising result.

sean_pedersen 5y ago

0.0?

ubavic 5y ago

There are some misleading sentences in this article. > The C stands for continuity and the superscript zero means the curves on the embedded surface have no derivatives, not even a first. This is not exactly true. C maps maybe have higher derivatives and maybe they don’t. Certainly, C isn’t a class of only continuous mappings that don’t have the first derivative. Also, the example with the letter U is bad (it tries to make a point, and it fails in doing so). A U-shaped curve can be perfectly smooth at all points. It would be “problematic” if we, for example, made the letter U by “glueing” a part of the circle and two segments.

SOFAYON 5y ago

>A key limitation of Cao and Inauen’s paper is that it requires embedding a shape into eight-dimensional space, rather than the three-dimensional space Gromov had in mind. With extra dimensions, the mathematicians gained more room to add twists, which made the problem easier. Mathematicians Identify Threshold at Which Shapes Give Way [In 8 Dimensions!]