> 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"?
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.
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> 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.
Kind of confusing naming them the same.
Milnor is a titan in his fields, so any conjecture he has made would be called Milnor's conjecture.
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
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.
So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?
Given your response, is it fair to say time as the 4th dimension is just a sci-fi concoction?
https://youtu.be/zwAD6dRSVyI?si=QC1s1JcMopGWEkUR
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.
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
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.
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