Poincare Conjecture and Ricci Flow | A Million Dollar Problem in Topology

what you're seeing now is called reach

flow

with surgery, it changes the curvature of the manifolds in four dimensional space. You may have heard of it because it was used to solve a

million

-

dollar

topology

problem

called the Poin Caray

conjecture

and by the end of this video, you will understand exactly what today's agenda is: understanding the

conjecture

and how Richi

flow

was used to solve it. Along the way, we'll learn some remanufactured geometry and theory of surgery and with these you'll have proof that Terren TOA called one of the most impressive recent achievements in modern mathematics.

The punctual car conjecture says this. Suppose you have a dough inside four dimensional space if the dough has no holes so it is not like this or like that, and if it is in one piece so it is not like this or like that, then you can mold The Blob into a sphere in formal terms, any three-way closed manifold that is connected and where each loop can be contracted to a point is homeomorphic to the three spheres that a manifold is. just a surface in a higher dimensional space a triple manifold is a surface that locally appears three-dimensional this is our connected blob path means you can connect any two points with a path this means you can squash any loop in space up to a point this means that you can mold The Blob into a sphere located within four-dimensional space.

Here's a question: who cares? It has legendary status because it sounds easy, for example, these are all terms you should learn in a first

topology

course, but no one has studied. able to decipher it for more than 100 years, even worse in all other dimensions, this

problem

was solved. The Nal 4 case was particularly difficult, so Michael Fredman won the 1986 Fields Medal for cracking it, but for some reason the Nal 3 case was impossible, it was solved. In 2002, the Russian mathematician Gia Perilman published three papers in the archive, they were rich in ideas, but Frugal with details now this will get really abstract very quickly, so keep FTI ranian geometry studies on size and curvature, which made with a device called a metric tensioner.

G, this thing takes a tangent vector to the manifold and assigns it a length, so maybe this arrow has length two and this arrow has length five. Note that the arrow has no intrinsic length, it is an abstract object, we are just giving it a length of its own. choosing the metric tensor has a nice intuitive picture imagine tying a shoelace through a point if we assign very small lengths to tangent vectors it is like pulling the shoelace if we assign really large links it is like relaxing the shoelace the metric tensioner It tells the manifold where to shrink and where to expand, but if you've noticed, we've also told the manifold how to curve and that's the amazing thing about the metric tensor when encoding size, it gives you curvature.

This brings us to the second player of remonia geometry, attainable curvature. this object tells you exactly how the manifold curves all in terms of G now full disclosure the exact formula for curvature the Richi curve is a little unwieldy which shouldn't be a surprise this curvature is quite complicated but the point is that if you know G, Know R now we are ready to apply this flow REI is a way to change the metric tensor over time so that the manifold becomes rounder. So how do we express Richi's flow concretely? I want you to focus on this region here, the REI flow is inflated.

It's like a balloon, by convention we say it has a negative reach curvature, so if the curvature is negative, the length increases. Now focus on this region. Here the Richi flow deflates it, so if the Richi curvature is positive, the length decreases. We can express this differently. G decreases. means that the derivative of G is negative G increases just means that the derivative of G is positive these two types always have opposite signs so we could guess an equation like this and that's it, that is the equation that describes the flow of REI as tradition, we put a two on this side because that's what the inventor of the REI flow did and it just stuck, but the meaning doesn't change.

With this we can understand something crucial to decipher the point. Damn, the REI flow crushes a sphere into nothing. A sphere has positive curvature everywhere, so the derivative of G is negative, so the metric keeps decreasing forever until it reaches zero. Perilman also showed that the opposite was true if the metric reached zero. The last shape you had must have been a sphere. Here's how we use this to solve the point. Damn it takes a random shape. manifold and assign it an arbitrary metric G and see how it changes with the flow REI if you can show that the metric will go to zero in finite time, that means the last shape it had must have been a sphere, but changing the metric doesn't change the underlying variety, so if you ended up with a sphere, you must have started with something homeomorphic for a sphere.

This argument is beautiful, but it has a problem that is seen in higher dimensions. You can have situations like this where a point on this neck is crushed to nothing. even before the manifold becomes a sphere, this is a problem if the REI flow removes parts of your manifold, it is changing the underlying set, it is a play on another Singularity, you might see a neck with a cap, a Rich flow could stretch and tighten this point. In nowhere there could also be more complicated obstacles like the brine giant Solon, where some of the points in the collector mysteriously disappear to deal with these singularities.

Perilman came up with a strategy. Perilman decided to manually remove the problem areas of the manifold and then glued the parts together. of spheres to cover the holes I had made this is how we used it to test the coray point play with your flow achievable with surgery and show that it will be extinguished now press the rewind button first and you will see the creation of spheres then you will see the creation of necks at each moment you will only see the creation of spheres and necks, but two spheres connected by a neck are just one sphere, so at each moment the array is a collection of spheres, but the array we started with was connected, so which must have been a sphere, say what you want, but this argument is simply beautiful and proves the point of attention conjecture, making it one of the best Tri.