# What mathematics did Perelman use to prove Poicare's conjecture? Do

• talisman2212
In summary, Perelman used Ricci flow, specifically in differential geometry and topology, to prove Poincaré's conjecture. For more information, a recommended book is "The Ricci Flow: An Introduction" by Bennett Chow and Dan Knopf. While some may criticize the use of Wikipedia, it is a helpful source for this topic for the majority of mathematicians.
talisman2212
What mathematics did Perelman use to prove Poicare's conjecture? Do you know any guide or any book?

Unless you are a specialist in differential geometry and topology, nobody's answer will be any more specific than the Wikipedia explanation. Perelman primarily used Ricci flow, as did Richard Hamilton before him. If you want a book, see
https://www.amazon.com/dp/0821843281/?tag=pfamazon01-20

N.B. Please don't be so quick to bash the other posters for recommending Wikipedia. They're trying to help, and it is the most helpful source for this topic for all but a tiny fraction of mathematicians. It doesn't mean that they're "not smart enough" to answer.

## 1. What is Poincare's conjecture and why is it important?

Poincare's conjecture is a famous unsolved problem in mathematics that states that any closed 3-dimensional manifold is topologically equivalent to a 3-sphere. It is important because it has implications in various fields such as topology, geometry, and physics.

## 2. Who is Grigori Perelman and why is he associated with the proof of Poincare's conjecture?

Grigori Perelman is a Russian mathematician who gained recognition for his proof of Poincare's conjecture in 2002. He is associated with the proof because he was the first to publish a complete solution to the problem, although it took several years for his proof to be fully accepted by the mathematical community.

## 3. What is the role of topology in Perelman's proof of Poincare's conjecture?

Topology is the branch of mathematics that studies the properties of geometric objects that are preserved under continuous deformations. Perelman's proof of Poincare's conjecture heavily relies on the techniques and concepts of topology, such as the use of Ricci flow and Thurston's Geometrization Conjecture.

## 4. How did Perelman use Ricci flow in his proof of Poincare's conjecture?

Ricci flow is a mathematical tool used to deform the metric of a manifold along its curvature. In his proof, Perelman showed that by applying Ricci flow to a 3-dimensional manifold, it can be transformed into a more simple and geometrically uniform manifold, which is a key step in proving Poincare's conjecture.

## 5. What are the remaining challenges in understanding Perelman's proof of Poincare's conjecture?

Although Perelman's proof has been accepted by the mathematical community, it is still considered to be a difficult and complex topic to fully understand. One of the challenges is the use of advanced mathematical concepts and techniques, such as Ricci flow and Thurston's Geometrization Conjecture, which may be unfamiliar to many mathematicians. Additionally, there are still some technical details that need to be fully understood and verified in order to completely grasp the proof.

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