> It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example M7 with Ric ≥ 0 such that π1(M) = Q/Z is infinitely generated.
It's like how there are so many things named after Euler. The joke goes that everything in math is named after the second person to discover it after Euler, but many things are named after him anyway.
Milnor is a titan in his fields, so any conjecture he has made would be called Milnor's conjecture.
> The joke goes that everything in math is named after the second person to discover it after Euler
This was one of the stories on my calculus book in college
I loved reading the brief explanations about the history of the different formulas and discoveries
In fact, reading those stories earned me a free “extra A” in class for remembering Rolle’s theorem. I only remembered it because Rolle’s story really caught my attention. After coming up with his theorem, he became a big critic of calculus, even though his theorem is one of its crucial building blocks
Is there a good layman’s explanation of higher dimensions as they relate to this type of problem? I’m trying to envision what that means, which is probably a wrong approach…
An n-dimensional space is just a collection of points, each defined uniquely by a set of n-numbers. The semantic meaning of those numbers doesn't really matter. It might be like actual physical space, but it could just as well be something like "time" and "the price of big macs". We have a bunch of mathematical operations that work well on 2 or 3 dimensional space that correlate nicely with our physical intuitions of 'curvature' and 'holes', and that still work perfectly well in more generalized forms in higher dimensions.
I'm not really sure it's that useful to try and visualize what it means on higher dimensions, to be honest.
It's not perfect but to get an idea of adding one more dimension on top of the three dimensions we can visualize is thinking of color as the 4th dimension. There's a game called 4D Maze created by a topolgist that's availble in iphone app store. The visualization is 3d but if you can imagine the colors taking up the same space (instead of being right next to each other in 3d space), it kinda works. At least, it's the closest I've ever come to feeling like I could visualize or understand an additional dimension.
Yeah, the "an n-dimensional vector is just a struct with n floats" way of thinking is great - until you actually want to apply geometrical operations in the vector space, such as calculating a distance or performing a rotation. Then you have a problem: You cannot visualise such a space and "pretending" to work in 2D/3D space is convenient but often extremely misleading.
So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?
“Just a struct” plus “measuring curvature and shapes” is where my mind goes into “must visualize this” mode. How does a struct have curvature/shape? Or is curvature overloaded here (with a technical math definition that is very different than the layman’s “surface of a sphere” mental model).
the technical math definition is a rigourous formulation that encapsulates exactly the same thing as what we mean when we say things are curved, but one that also extends far more generally into contexts where our old intuition fails.
The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.
No, that's not exactly a sci-fi concoction. In special and general relativity, there are three dimensions for space and one dimension for time, and this is not something that is of "incidental" importance to special / general relativity, it's a pretty essential shift in perspective to these theories to think of the universe as (curved) four-dimensional spacetime.
But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).
Contextually of dimension even exists within a specific scientific theory. In relativity, the direction you call time might contain some component of the direction I call space. This implies notions like simultaneity are not well defined in a universal context.
No, 4D spacetime is a real thing in physics, which explains things like time dilation and the speed of light. But sci-fi does tend to abuse the term "dimension" for other ideas that are not scientific.
yeah, nobody can visualize it. it's something you just get used to after a while.
there's an old joke about a mathematician teaching an engineer about thirteen-dimensional spaces. "What do you think," the mathematician asks. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?"
"Well, it's not so hard. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13."
Three Blue, One Brown has a decent video which perhaps helps get a handle on how one can use higher-dimensional spaces, without needing to wrap you head around trying to envision a four-dimensional cube, or similar.
From what I've heard, a fair number of the mathematicians doing research on 4+-dimensional things seem to have developed very good intuition about them, and okay-ish ability to "visualize" them. Those abilities falls off as you add dimensions, or try applying them to more-complex shapes...which is hardly surprising, considering how an average person's ability to intuit and visualize (in dimensions 0 through 3) falls off when complexity and dimensions are added.
Yes actually, I've been binging a lot of General Relativity content lately by coincidence and can say the Dialect YouTube channel has been the best resource for describing this. I'm not an expert so I cannot speak to its accuracy but it seems sound.
In particular the video "Conceptualizing the Christoffel Symbols". Also look at content on the Metric Tensor
Additionally, there is content from other sources (albeit less produced) on describing projective geometry which is also related
Everybody is giving advice on how to imagine and intuit about integer dimensions beyond 3.
But what about fractional dimensions (not to be confused with fractal dimensions)? Any advice about reasoning about the geometry of lets say something 0.6309297535... dimensional? It seems so easy, I mean it is somewhere between 0 and 1 dimensions, both of which have trivial geometric interpretations.
Closest I could think of is doing augmentations into the next highest integer dimension. That would be similar to how we often use projections to lower integer dimensions to think about higher integer dimensions, but in reverse.
And yes, fractional dimensions do exist, just like fractional derivatives or fractional Fourier transform, etc.
Maybe not exactly what you are describing, but I recently did some layman research on "Strange Attractors" and chaos theory, which covers very similar topics. I cannot summarize here, but it's a neat rabbit hole to go down
1. create a copy of the point and move it a fixed distance along a specific direction (which we will call the X-axis) and join the two points together -- this is a 1D line;
2. create a copy of the line and move it along a specific direction that is orthogonal (at 90 degrees to) to the direction the line is facing (which we will call the Y-axis) and join the two end points together -- this is a 2D square;
3. create a copy of the square and move it along a specific direction that is orthogonal (at 90 degrees to) to the other (X and Y) directions (which we will call the Z-axis) and join the four end points together -- this is a 3D cube;
4. for 4D and higher dimensions you can repeat this process to form other hypercubes [1]. The 4D version is a tesseract [2] on whih you can see this construction.
The general approach of visualising these is to use a projection [3] in a way similar to how a cube is displayed on a 2D screen or image. The idea is to cast a shadow to the next dimension down. For a hypercube you can project this to 3D space and then to 2D space.
Dimensions are typically defined in terms of unit vectors. These are vectors pointing in the X, Y, Z, ... directions with a length of 1. I.e. all coordinates are 0 except for the direction which is 1. For 4 dimensions they will have the values x̂ = (1,0,0,0), ŷ = (0,1,0,0), etc. Thus, you can express coordinates as multiples of these unit vectors. (This is similar to the x + iy notation for complex numbers, where 1 is the real unit vector and i is the imaginary unit vector.)
A hypersphere is an n-dimensional object where the points on the surface are a fixed distance (radius) away from the hypersphere's origin. This is typically defined as sum(s_n^2) = 0 -- for a circle (2D) this becomes x^2 + y^2 = 0; for a sphere (3D) this becomes x^2 + y^2 + z^2 = 0.
A manifold is just a generalized closed n-dimensional surface, such as the surface of a cube, circle, donut, or other object [4]. It is defined as a set (collection) of the points on that surface. These points can be defined as an equation, such as the equation of a hypersphere, or more generally. For example, you could define each face of a hypercube separately.
Link to the paper: "Fundamental Groups and the Milnor Conjecture", https://arxiv.org/abs/2303.15347
> It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. The main result of this paper is a counterexample, which provides an example M7 with Ric ≥ 0 such that π1(M) = Q/Z is infinitely generated.
There's another Milnor conjecture: https://en.wikipedia.org/wiki/Milnor_conjecture
Kind of confusing naming them the same.
It's like how there are so many things named after Euler. The joke goes that everything in math is named after the second person to discover it after Euler, but many things are named after him anyway.
Milnor is a titan in his fields, so any conjecture he has made would be called Milnor's conjecture.
> The joke goes that everything in math is named after the second person to discover it after Euler
This was one of the stories on my calculus book in college
I loved reading the brief explanations about the history of the different formulas and discoveries
In fact, reading those stories earned me a free “extra A” in class for remembering Rolle’s theorem. I only remembered it because Rolle’s story really caught my attention. After coming up with his theorem, he became a big critic of calculus, even though his theorem is one of its crucial building blocks
Is there a good layman’s explanation of higher dimensions as they relate to this type of problem? I’m trying to envision what that means, which is probably a wrong approach…
Higher dimensions in general?
An n-dimensional space is just a collection of points, each defined uniquely by a set of n-numbers. The semantic meaning of those numbers doesn't really matter. It might be like actual physical space, but it could just as well be something like "time" and "the price of big macs". We have a bunch of mathematical operations that work well on 2 or 3 dimensional space that correlate nicely with our physical intuitions of 'curvature' and 'holes', and that still work perfectly well in more generalized forms in higher dimensions.
I'm not really sure it's that useful to try and visualize what it means on higher dimensions, to be honest.
It's not perfect but to get an idea of adding one more dimension on top of the three dimensions we can visualize is thinking of color as the 4th dimension. There's a game called 4D Maze created by a topolgist that's availble in iphone app store. The visualization is 3d but if you can imagine the colors taking up the same space (instead of being right next to each other in 3d space), it kinda works. At least, it's the closest I've ever come to feeling like I could visualize or understand an additional dimension.
Yeah, the "an n-dimensional vector is just a struct with n floats" way of thinking is great - until you actually want to apply geometrical operations in the vector space, such as calculating a distance or performing a rotation. Then you have a problem: You cannot visualise such a space and "pretending" to work in 2D/3D space is convenient but often extremely misleading.
So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?
“Just a struct” plus “measuring curvature and shapes” is where my mind goes into “must visualize this” mode. How does a struct have curvature/shape? Or is curvature overloaded here (with a technical math definition that is very different than the layman’s “surface of a sphere” mental model).
the technical math definition is a rigourous formulation that encapsulates exactly the same thing as what we mean when we say things are curved, but one that also extends far more generally into contexts where our old intuition fails.
The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.
Yeah, the visualization is where I get hung up. Sounds like I need to stop that.
Given your response, is it fair to say time as the 4th dimension is just a sci-fi concoction?
No, that's not exactly a sci-fi concoction. In special and general relativity, there are three dimensions for space and one dimension for time, and this is not something that is of "incidental" importance to special / general relativity, it's a pretty essential shift in perspective to these theories to think of the universe as (curved) four-dimensional spacetime.
But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).
Contextually of dimension even exists within a specific scientific theory. In relativity, the direction you call time might contain some component of the direction I call space. This implies notions like simultaneity are not well defined in a universal context.
No, 4D spacetime is a real thing in physics, which explains things like time dilation and the speed of light. But sci-fi does tend to abuse the term "dimension" for other ideas that are not scientific.
yeah, nobody can visualize it. it's something you just get used to after a while.
there's an old joke about a mathematician teaching an engineer about thirteen-dimensional spaces. "What do you think," the mathematician asks. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?"
"Well, it's not so hard. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13."
Geoffrey Hinton on visualizing higher dimensions:
"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it."
Three Blue, One Brown has a decent video which perhaps helps get a handle on how one can use higher-dimensional spaces, without needing to wrap you head around trying to envision a four-dimensional cube, or similar.
https://youtu.be/zwAD6dRSVyI?si=QC1s1JcMopGWEkUR
A game called 4D Golf came out recently if you want to try interacting with the physics of minigolf in 4 spatial dimensions.
About higher dimensions and envisioning:
From what I've heard, a fair number of the mathematicians doing research on 4+-dimensional things seem to have developed very good intuition about them, and okay-ish ability to "visualize" them. Those abilities falls off as you add dimensions, or try applying them to more-complex shapes...which is hardly surprising, considering how an average person's ability to intuit and visualize (in dimensions 0 through 3) falls off when complexity and dimensions are added.
Yes actually, I've been binging a lot of General Relativity content lately by coincidence and can say the Dialect YouTube channel has been the best resource for describing this. I'm not an expert so I cannot speak to its accuracy but it seems sound.
In particular the video "Conceptualizing the Christoffel Symbols". Also look at content on the Metric Tensor
Additionally, there is content from other sources (albeit less produced) on describing projective geometry which is also related
Everybody is giving advice on how to imagine and intuit about integer dimensions beyond 3.
But what about fractional dimensions (not to be confused with fractal dimensions)? Any advice about reasoning about the geometry of lets say something 0.6309297535... dimensional? It seems so easy, I mean it is somewhere between 0 and 1 dimensions, both of which have trivial geometric interpretations.
Closest I could think of is doing augmentations into the next highest integer dimension. That would be similar to how we often use projections to lower integer dimensions to think about higher integer dimensions, but in reverse.
And yes, fractional dimensions do exist, just like fractional derivatives or fractional Fourier transform, etc.
Maybe not exactly what you are describing, but I recently did some layman research on "Strange Attractors" and chaos theory, which covers very similar topics. I cannot summarize here, but it's a neat rabbit hole to go down
Starting with a 0 dimensional (0D) point:
1. create a copy of the point and move it a fixed distance along a specific direction (which we will call the X-axis) and join the two points together -- this is a 1D line;
2. create a copy of the line and move it along a specific direction that is orthogonal (at 90 degrees to) to the direction the line is facing (which we will call the Y-axis) and join the two end points together -- this is a 2D square;
3. create a copy of the square and move it along a specific direction that is orthogonal (at 90 degrees to) to the other (X and Y) directions (which we will call the Z-axis) and join the four end points together -- this is a 3D cube;
4. for 4D and higher dimensions you can repeat this process to form other hypercubes [1]. The 4D version is a tesseract [2] on whih you can see this construction.
The general approach of visualising these is to use a projection [3] in a way similar to how a cube is displayed on a 2D screen or image. The idea is to cast a shadow to the next dimension down. For a hypercube you can project this to 3D space and then to 2D space.
Dimensions are typically defined in terms of unit vectors. These are vectors pointing in the X, Y, Z, ... directions with a length of 1. I.e. all coordinates are 0 except for the direction which is 1. For 4 dimensions they will have the values x̂ = (1,0,0,0), ŷ = (0,1,0,0), etc. Thus, you can express coordinates as multiples of these unit vectors. (This is similar to the x + iy notation for complex numbers, where 1 is the real unit vector and i is the imaginary unit vector.)
A hypersphere is an n-dimensional object where the points on the surface are a fixed distance (radius) away from the hypersphere's origin. This is typically defined as sum(s_n^2) = 0 -- for a circle (2D) this becomes x^2 + y^2 = 0; for a sphere (3D) this becomes x^2 + y^2 + z^2 = 0.
A manifold is just a generalized closed n-dimensional surface, such as the surface of a cube, circle, donut, or other object [4]. It is defined as a set (collection) of the points on that surface. These points can be defined as an equation, such as the equation of a hypersphere, or more generally. For example, you could define each face of a hypercube separately.
[1] https://en.wikipedia.org/wiki/Hypercube
[2] https://en.wikipedia.org/wiki/Tesseract
[3] https://en.wikipedia.org/wiki/Projective_geometry
[4] https://en.wikipedia.org/wiki/Manifold
It's about Milnor’s conjecture.
Still waiting with bated breath for their geometric Langlands article ;)