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

[Music] what you're looking at right now is called Ricci flow with surgery it changes the curvature of manifolds in four-dimensional space you may have heard of it because it was used to crack a 1 million dollar problem in topology called the Poincare a conjecture and by the end of this video you'll understand exactly what it is the agenda for today is to understand the conjecture and how Ricci flow was used to solve it along the way we'll pick up some romani and geometry and surgery theory and with these we'll have the proof that Terence Tao called one of the most impressive recent achievements of modern mathematics [Music] the Poincare conjecture says this suppose you have a blob sitting inside four dimensional space if the blob has no holes so not like this or like this and if it's in one piece so not like this or like this then you can mold the blob into a sphere in formal terms any closed three manifold that is path connected and where every loop is contractible to a point is homeomorphic to the three sphere a manifold is just a surface in higher dimensional space a three manifold is a surface that locally looks three-dimensional this is our blob path connected means that you can connect any two points with a path this means that you can squish any loop in the space to a point this means that you can mold the blob into a sphere sitting inside four dimensional space here's a question who cares well it has legendary status because it sounds easy for example these are all terms that you should learn in a first course in topology but nobody's been able to crack it for over a hundred years were still in every other dimension this problem was solved the N equals four case was particularly hard so Michael Friedman won the 1986 Fields Medal for cracking it but for some reason the N equals three case was impossible it was solved in 2002 Russian mathematician Grisha Perelman posted three papers on the archive they were rich in ideas but frugal with details now this is going to get really abstract really fast so hold on tight [Music] ramani and geometry studies size and curvature it does so with a gadget called the metric tensor G this thing takes a tangent vector to the manifold and assigns it a length so maybe this arrow has length 2 and this arrow has length 5 note the arrow has no intrinsic length it's an abstract object we're just giving it a length of our own choosing the metric tensor has a nice intuitive picture imagine tying a shoelace through a point if we assign really small lengths to tangent vectors it's like pulling in the Shoeless if we assign really large legs it's like relaxing the shoelace the metric tensor tells the manifold where to shrink and where to expand but if you've noticed we've also pulled the manifold how to curve and that's the amazing thing about the metric tensor by encoding size it gives you perversion this brings us to the second player of Romani in geometry the Ricci curvature this object tells you exactly how the manifold curves all in terms of G now full disclosure the exact formula for the Ricci curvature is a little unwieldy that shouldn't be a surprise this curvature is quite complicated but the point is if you know G you know R now we're ready to apply these Ricci flow is a way of changing the metric tensor over time so that the manifold becomes rounder so how do we express Ricci flow concretely I want you to focus on this region over here the Ricci flow inflates it like a balloon by convention we say that it has negative Ricci curvature so if the curvature is negative the length increases now focus on this region here Ricci flow deflates it so if the Ricci curvature is positive the length decreases we can phrase this differently G decreases means the derivative of G is negative G increases just means the derivative of G is positive these two guys always have opposite signs so we might get an equation like this and that's it that is the equation describing Ricci flow as tradition we put a 2 on this side because that's what the inventor of Ricci flow did and it just kind of stuck but the meaning is unchanged with this we can understand something crucial to cracking the Pointer a Ricci flow squishes a sphere to nothingness a sphere has positive curvature everywhere so the derivative of G is negative so the net trick keeps decreasing forever until it hits zero Perelman also showed that the opposite was true if the metric went to zero the last shape you had must have been a sphere here's how we use this to solve the point array take a random manifold and give it an arbitrary metric G and see how it changes with Ricci flow if you can prove the metric will go to zero in finite time that means the last shape you had must have been a sphere but changing the metric doesn't change the underlying manifold so if you ended up with a sphere you must have started with something homeomorphic to a sphere this argument is beautiful but it hits a snag you see in higher dimensions you can have situations like this where a point in this neck gets squished to nothing even before the manifold becomes a sphere this is a problem if Ricci flow deletes parts of your manifold you're changing the underlying set it's game over another singularity you might see a neck with a cap a Ricci flow could stretch this out and squeeze this point into nothingness you could also have more complicated snags like the Bryant soliton where some of the points in the manifold mysteriously vanish to deal with these singularities Perelman came up with a strategy [Music] Perelman decided to manually remove the problem areas of the manifold he then glued pieces of spheres to cover the holes that he had made here's how we used it to prove the point correct play out to your Ricci flow with surgery and prove that it'll go extinct now hit the rewind button first you'll see the creation of spheres then you'll see the creation of necks at each time you only see the creation of spheres and necks but two spheres connected by a neck is just a sphere so at each time the manifold is a collection of spheres but the manifold we started off with was connected so it must have been a sphere say what you want but this argument is just beautiful and improves the pointer a conjecture making it one of the greatest triumphs of modern mathematics [Music] you

Channel: Aleph 0

Views: 170,029

Rating: 4.9235234 out of 5

Keywords: mathematics, math, problem, topology, geometry, differential geometry, poincare conjecture, millennium prize problem, reimann hypothesis, ricci flow, P vs NP, curvature, million dollar

Id: PwRl5W-whTs

Length: 8min 27sec (507 seconds)

Published: Wed Jun 24 2020