Keeping in mind I've only done basic statistics and calculus as part of my ongoing medical degree. I've also done high school level (GCSE) maths.
I'm currently studying the Princeton's companion to mathematics in my free time, I imagine knowledge from this book would be no where near enough to understand the Poincaré Conjecture.
Thanks for any input :)
I have very little understanding and I need to use it in a debate case. Any ideas how to dumb it down down?
In the last episode of season 4, minute 13:40, there is a "to do list" by Tahani, with some stuff already ruled out; among these last ones there is "solve the Poincaré conjecture". But, the, Poincaré conjecture was solved in 2002 (https://it.m.wikipedia.org/wiki/Congettura_di_Poincar%C3%A9). How is this possibile? Do you think this has some meaning or it's just a mistake?
I have no math background other than school and tiny bit of engineering info from my firends. I have tried and tried to figure out what the Poincaré conjecture is about , and i watched the Grigori Perelman documentary (which is awesome) and a lot of youtube videos like numberphile ones and such. I still can grasp the idea of it , i know it has something to do with holes and all dimentions except the third one can't have an object with holes or something like that. Please can someone explain it to me ?
edit thank you for all the replies , so many intelligent people ready to share their time and help me out, i appreciate it , and all of you gave me a way better understanding of the problem .
So, I ask this as a layman who is familiar with the big unanswered questions in math but doesn't pretend to have any expertise in them.
For most of the "hard questions" that I have come across, I just sort of facially understand why are they are hard. Like, if all the non-trivial zeroes to the Zeta function are of the form 1/2+ai, that's not remotely intuitively obvious and I would have no idea how to prove it. Seems hard. Same for something like Fermat's Last Theorem - the question is simple, but the answer is both not obviously true and I have no clue how I'd prove it.
But to my layman's brain, the Poincare Conjecture seems sort of...obvious? I don't mean to sound like an idiot or insufferably arrogant here, since obviously it's not obvious - I'm just trying to get a better intuitive understanding of why.
If you take the 2D case, it just seems obvious that any simply connected line without any holes or overlap can be pushed and pulled and deformed until you get a circle. Same for a 3D sphere - if you have a shell without any holes and which never intersects itself, of course it can be molded into a sphere. So why would the 4D 3-sphere be not obviously the same?
I guess the two ways I can see I'm going wrong here are (i) even for the 2D or 3D case, it's actually not obvious and I'm missing some complications and/or (ii) the 4D case (which is what I understand the Poincare Conjecture to really be about) is just immensely more complicated than the others.
Can someone help me understand the difficulty of the problem?
Two recent examples of advanced solved math problems and their proofs' method are Andrew Wiles' proof of Fermat's Last Theorem, using advanced applications of elliptic curves and highly specialized theorems; and Grigori Perelman's proof of the Poincaré Conjecture, using the Riemannian Metric, modifications of Ricci Flow, and (apparently) not-too-exotic applications of manifolds.
My summaries above of the proofs' main methods are probably too general. I'm wondering about the topic in general, so here are a few questions I've sussed out to try to get at the core of what I'm trying to learn more about:
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Can these, or other highly advanced math problems, be solved using highly different methods and approaches? (I would, of course, still expect a proof of a topology problem to be achieved using primarily tools from the field of topology.)
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Are the problems too advanced and specialized for highly different proofs to be meaningfully produced? In other words, is there a limit as to how "different" such alternate proofs can end up being?
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Is it ever useful to even try to tackle these kinds of problems from two highly unrelated directions?
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And catch-all: Is there anything else fundamental to this issue that I overlooked or that would be interesting to know?
Thanks for all of your detailed insight!
Hey guys. So I wanted to know if anyone can tell me what notions I should be familiar with to understand the proof of this conjecture. (I already have a background in calculus, algebra, geometry and statistics).
Context: I know next to nothing about the Poincaré conjecture proof or the more advanced results of general relativity (i'm a grad student in PDEs). But while quickly browsing through some general relativity stuff and the Perelman's proof (just gazing at it), I noticed some similarities/themes in the two (Ricci flows, metric tensors, deformation manifolds....) . Am I way out in left field here or is there some connection? Looking for general comments/discussion about the two topics. Thanks.
I tried reading about it and couldnt understand. Thanks in advance
Been reading about the Poincaré Conjecture, the only million dollar millennium problem to be solved.
I think I sorta get it, but an ELI5 on it would really help!
A simplified explanation of 3-spheres/n-spheres would also be greatly appreciated!
This is obviously a hard ass probm to wrestle with, as well as being really recent, but does anyone with a better grasp on topology have an idea of what, if any, proof techniceques Grigori Perelman used would work for the general case? (I'd guess Ricci flow would still be hella usefull, but non-linear PDEs are anything but easy to work with)
Video I'm talking about: https://www.youtube.com/watch?v=-6g3ZcmjJ7k
Seems like the video was made before the proof was published but wondering if they're related? Sorry I'm not really a math person, I just lurk here.
Just watched a few videos and read a few articles about it and was curious
