Ricci Flow - Numberphile
video description
An interesting example would be looking at another person, wearing a colourful t-shirt. You don't see generated light, but reflected light. If we think of light as a particle photon, the reflected photon or reflected light, would reflect this line of photons in a very specific direction. Since you can see this particular line of particle photons, it would mean you and only you could ever have received/seen that photon, that light and that energy.
Date: 2022-04-08
Related videos
Comments and reviews: 9
sayunts
Ricci flow is described by the evolutionary equation with the metric of a manifold changing over the time on the left side and the Ricci tensor on the right side, multiplied by factor 2.
Even when curvature is varying fron point to point on the maniflod (say, surface) it is not a flow
yet. Flow begins when the metric tensor (g, that defines the distance between two point on the manifold, starts changing with time. You can imagine like the surface experiences deformations changing its shape from time to time. In this case the curvature, which is defined by Riccci tensor, is varying with time - increasing at one point and decreasing at another point. Physically it reminds you a flow of heat energy in the volume from place to place.
reply
Ricci flow is described by the evolutionary equation with the metric of a manifold changing over the time on the left side and the Ricci tensor on the right side, multiplied by factor 2.
Even when curvature is varying fron point to point on the maniflod (say, surface) it is not a flow
yet. Flow begins when the metric tensor (g, that defines the distance between two point on the manifold, starts changing with time. You can imagine like the surface experiences deformations changing its shape from time to time. In this case the curvature, which is defined by Riccci tensor, is varying with time - increasing at one point and decreasing at another point. Physically it reminds you a flow of heat energy in the volume from place to place.
reply
Tetraedri_
Huh, I'm surprised how he choce to explain Riemannian geometry. It was quite confusing and, at least to my taste, wasn't very insightful.
Intuitively, I like to think Riemannian geometry to be a description of the geometry of (d-dimensional) surface as experienced by a creature living in there, with their sences restricted to observe only the surface. For example, if we take a 2D surface and imagine a 2D space ship traversing the surface (think of e. g. Asteroids-game on a torus, then the metric tells how the space ship experiences the geometry of the surface around it. Thus, metric is an intrinsic description of the geometry of a surface, so the notion of -space where and how it lives- (i. e. embedding) is abstracted away.
reply
Huh, I'm surprised how he choce to explain Riemannian geometry. It was quite confusing and, at least to my taste, wasn't very insightful.
Intuitively, I like to think Riemannian geometry to be a description of the geometry of (d-dimensional) surface as experienced by a creature living in there, with their sences restricted to observe only the surface. For example, if we take a 2D surface and imagine a 2D space ship traversing the surface (think of e. g. Asteroids-game on a torus, then the metric tells how the space ship experiences the geometry of the surface around it. Thus, metric is an intrinsic description of the geometry of a surface, so the notion of -space where and how it lives- (i. e. embedding) is abstracted away.
reply
Sankaravelayudhan
The fluidity of water becoming a Boson. condensate or Fermion condensate forming a fluidity at the topological surface forming from a critical point in depth forming a typical entropy of phase say if water condensate at critical point at depth the density suddenly decreases and floats to the top as ice blocks. Perhaps a fluidity over which ice blocks float may go up or down according to the fluid curvature in planets forming Bose Einstein fluidity. When the fluid comes out to the surface a Ricci flow may be observed based on the topological curvature produced.
Sankaravelayudhan. Nandakumar.
reply
The fluidity of water becoming a Boson. condensate or Fermion condensate forming a fluidity at the topological surface forming from a critical point in depth forming a typical entropy of phase say if water condensate at critical point at depth the density suddenly decreases and floats to the top as ice blocks. Perhaps a fluidity over which ice blocks float may go up or down according to the fluid curvature in planets forming Bose Einstein fluidity. When the fluid comes out to the surface a Ricci flow may be observed based on the topological curvature produced.
Sankaravelayudhan. Nandakumar.
reply
nihal
Okay layman here. If I am getting this right, if you have an object being pressed insided/outside with force acting perpendicularly on each point. then you have them forming circle or sphere ( like we see in nature with planets, bubbles, etc. But if we have the initial shape of it being with varying curvates ( notably the ones with large curvature) you would have bottlenecks collapsing into singularity. So the ricci flow is to find how the surfaces aroundd this bottle neck can be isolated. because if not the entire thing collapses into singularity.
reply
Okay layman here. If I am getting this right, if you have an object being pressed insided/outside with force acting perpendicularly on each point. then you have them forming circle or sphere ( like we see in nature with planets, bubbles, etc. But if we have the initial shape of it being with varying curvates ( notably the ones with large curvature) you would have bottlenecks collapsing into singularity. So the ricci flow is to find how the surfaces aroundd this bottle neck can be isolated. because if not the entire thing collapses into singularity.
reply
Antonis
What about when the shape shrinks towards the middle pressure increases and there are certain aspects we need to consider. One of them is if we come to a shrinking point that would let go by natural pressure force, we might not only get to the original point but we could have an expansion of the shape but also possible a separation of it in larger or smaller shape particles. Changing the flow of shapes, we also need to take into consideration the pressure that is exercised in Pascal (Pa) or Newton /m 2. Push or pull.
reply
What about when the shape shrinks towards the middle pressure increases and there are certain aspects we need to consider. One of them is if we come to a shrinking point that would let go by natural pressure force, we might not only get to the original point but we could have an expansion of the shape but also possible a separation of it in larger or smaller shape particles. Changing the flow of shapes, we also need to take into consideration the pressure that is exercised in Pascal (Pa) or Newton /m 2. Push or pull.
reply
Frank
My conclusion: Ricci flow is a specific prescription to alter a bunch of functions g_ij (the -metric-) defined on a connected set of points (-manifold-. This transformation is guided by a quantity (-curvature-) which can be calculated at each point from those -metric- functions g_ij. A somewhat similar prescription would be -curvature flow-, here demonstrated for special cases where the point sets are bubbles in 3D Euklidian space or closed loops in 2D and -curvature- would be the visible pointiness at a given point.
reply
My conclusion: Ricci flow is a specific prescription to alter a bunch of functions g_ij (the -metric-) defined on a connected set of points (-manifold-. This transformation is guided by a quantity (-curvature-) which can be calculated at each point from those -metric- functions g_ij. A somewhat similar prescription would be -curvature flow-, here demonstrated for special cases where the point sets are bubbles in 3D Euklidian space or closed loops in 2D and -curvature- would be the visible pointiness at a given point.
reply
Peter
this all makes sense up until the point where no one asks why is it OK to have the Ricci flow generate a singularity. We're trying to follow how Perelman's proof uses Ricci flow to show that certain initial conditions are always smoothly transformable to a plain old sphere without any holes, but introducing singularities doesn't seem sound, why are we so sure that the singularity isn't now covering up or hiding a hole that we were trying to prove doesn't exist?
reply
this all makes sense up until the point where no one asks why is it OK to have the Ricci flow generate a singularity. We're trying to follow how Perelman's proof uses Ricci flow to show that certain initial conditions are always smoothly transformable to a plain old sphere without any holes, but introducing singularities doesn't seem sound, why are we so sure that the singularity isn't now covering up or hiding a hole that we were trying to prove doesn't exist?
reply
d4slaimless
So at 1: 48 the curvature is very big, so this points moves very fast (transforms faster than point with low curvature. At 4: 35 the curvature getting arbitrary big, it forms singularity and stops changing. Means transforms very fast to singularity (arbitrary small point?
At 10: 00 I was thinking that I understand it more or less. But then I kind of got lost. Nice try though, at least I understood something )
reply
So at 1: 48 the curvature is very big, so this points moves very fast (transforms faster than point with low curvature. At 4: 35 the curvature getting arbitrary big, it forms singularity and stops changing. Means transforms very fast to singularity (arbitrary small point?
At 10: 00 I was thinking that I understand it more or less. But then I kind of got lost. Nice try though, at least I understood something )
reply
Kurtlane
I am sorry, but no. Hourglass shape, as long as it is continuous and differentiable, will turn into a sphere. The tight points (circle) in the middle will go out the most, the points around them will also go out, there will be circles on both sides that will stay the same, and beyond that everything will go in. I can picture it in my mind.
reply
I am sorry, but no. Hourglass shape, as long as it is continuous and differentiable, will turn into a sphere. The tight points (circle) in the middle will go out the most, the points around them will also go out, there will be circles on both sides that will stay the same, and beyond that everything will go in. I can picture it in my mind.
reply
Add a review, comment
Other channel videos
