Geometry and The Poincare Conjecture

While the 19th century witnessed a near complete understanding of 1 and 2 dimensional manifolds, much progress during the 20th century in 3 dimensions was guided by a conjecture of Poincare, first formulated in 1904. Poincare's conjecture — essentially that simple 3 dimensional spaces can be probed effectively using 1-dimensional loops — proved much more difficult than originally hoped, remaining unsolved for nearly 100 years. Following a century of work, its eventual resolution by Perelman in 2002 provided a new and powerful tool — called Geometrization — to the study of all 3-dimensional spaces. My goal in this talk is to give an overview of this exciting story lying at the heart of modern topology, from what was asked to what the mathematical community has learned.