AMR Colloquium

An introductory lecture by Bernhard Riemann explains the background, history, and the  context of the colloquium talk.

The Poincare Conjecture dates back to 1905 when Poincare asked if a 3-dimensional manifold  with trivial homology groups was homeomorphic to a 3-sphere.  He himself realized a restatement was needed, as the Poincare homology sphere gives a contradiction to the original statement. … 

Suggested background reading to follow the  ideas of the lecture.

Online videos related to the lecture.

A variety of approaches to the conjecture.

Some open problems connected to the Poincare Conjecture

How this solution affects topology.  What are the likely directions going forward.

Questions submitted to the speaker after the lecture and responses by the speaker.

Question: Why is dimension four different from dimension 5, or 50?

Answer: A key step in the process is to slide around curves that give instructions on how to build a manifold.  These curves trace out 2-dimensional surfaces as they move.  Pairs of such surfaces are disjoint in dimensions above 4, but can have irremovable intersections in dimension 4 and below.  Thus things change drastically when moving from dimension 4 to dimension 5.  For more details see [references].