Graduate What is the significance of the Poincaré conjecture?

Click For Summary
Perelman's proof of the Poincaré conjecture is significant as it contributes to the classification of all 3-manifolds. His work demonstrates the more general geometrization conjecture for closed 3-manifolds, encompassing the Poincaré conjecture as a specific instance. This achievement highlights the depth of Perelman's mathematical insight and expertise. The implications of his proof extend beyond the conjecture itself, influencing the broader field of topology. Overall, Perelman's contributions mark a pivotal moment in mathematical history.
donglepuss
Messages
17
Reaction score
4
TL;DR
What is the significance of the Poincaré conjecture?
Namely, what does Perelman’s proof of it imply?
 
Mathematics news on Phys.org
Wikipedia said:
The proof of the Poincaré conjecture is an important contribution to the classification of all 3-manifolds. This is because Perelman actually proves the more general geometrization conjecture over closed 3-manifolds, which includes the Poincaré conjecture as a special case.
And it proved that Perelman is a very special person.
 
Reply
  • Haha
  • Informative
Likes pinball1970, jedishrfu and mcastillo356

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Graduate Question about the Poincaré conjecture
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Questions regarding Kurepa's Conjecture
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proving Goldbach's Conjecture for Math Proofs
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Degenerate Triangles: Questions & Poincaré Conjecture
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Rewriting the Toeplitz Conjecture
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Breakthrough in 3n+1 Problem: Collatz Conjecture
  • · Replies 7 ·
Replies
7
Views
5K
High School Confusion with the basics of Topology (Poincare conjecture)
  • · Replies 2 ·
Replies
2
Views
3K
Undergrad Monumental Proof Settles Geometric Langlands Conjecture
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Proving Goldbach's conjecture by induction (or why it doesn't seem to work)
  • · Replies 10 ·
Replies
10
Views
3K
Graduate Can this proposition be proved in the Collatz conjecture?
  • · Replies 9 ·
Replies
9
Views
2K