Mathematisches Kolloquium

Zusammenfassung: The solution of the Poincaré Conjecture is the result of a fruitful interplay between topology, geometry and analysis. It was originally formulated in 1904 as a concise topological characterization of the three-dimensional sphere. After surviving various interesting attempts of proof it was subsumed in the late 70s by Thurston's Geometrization Conjecture which provides a rough classification scheme for the three-dimensional manifolds.

The Ricci flow introduced by R. Hamilton in the early 80s turned out to be a useful analytic tool in carrying out this very general program. It is a geometric evolution equation that gives a way of deforming metrics and improving their geometric properties. The crucial issue is to obtain good geometric control on the possible degenerations of metrics under the Ricci flow and to relate them to the topological structure of the underlying manifold.

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