Impact of Poincaré conjecture on mathematics?

  • #1
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In 2006, the mathematically rigorous proof of the Poincaré conjecture was completely excepted. The Poincaré conjecture was put forward by Henri Poincaré in the early 1900's. It is a theorem about the characterization of the 3D sphere amongst 3D manifolds. It was considered one of the most important things to prove for the millennium. And thus it was one of the clay mathematics institute Millennium problems, they would reward $1,000,000 to the person(s) who give a proof/disproof of the problem, there is 7 problems and in total $7,000,000 dollars up for grabs!

Generally the Poincaré conjecture is revolved around Topology, which deals with spaces. Apparently the Poincaré conjecture fixes up singularities in the dimensions e.t.c.

My question is, what are all the ramifications for the solution of the Poincaré conjecture, i.e. does it give proof or rise to new mathematics like quantum cohomology (Apparently, one of the other problems, call the Yang-mills existence and mass gap gives rise to Quantum cohomology, but Yang-mills theory has to have a mathematically rigorous background before quantum cohomology can be taken seriously)?

Also, What are the physical implication of the proof of Poincaré conjecture i.e. as i mentioned previously, this proof fixes the singularities in topology, so topologically speaking, what are the ramifications for blackholes and the big bang?

All you well educated opinions are well appreciated, as i am just a layman
 

Answers and Replies

  • #2
HallsofIvy
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So you are asking about applications to physics, not mathematics? You might do better asking in the physics forum.
 
  • #3
mathwonk
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i have not kept up with the work on the poincare problem, but in general i suspect it takes a while for the mathematical ramifications of a great proof to arise. often the ramifications in mathematics arise from further applications of the ideas that came into being to solve the problem.
 

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