The nineteenth century saw the development of Geometry in 2
  dimensions from the pioneering work of Abel and Riemann to its full
  flowering in the hands of Klein and Poincare. The first half of the
  twentieth century moved geometry into higher dimensions, with the
  emphasis on topology initiated by Poincare and developed by
  Lefschetz and Hodge. Towards the end of the century interest focused
  on the lower dimensions of 3 and 4, stimulated in great part by
  ideas from physics.

  The great achievement of Perelman, following on from the Thurston
  programme, closes a chapter in dimension 3 with the affirmative
  answer to the fundamental problem identified by Poincare. But there
  is still much to learn in 3 dimensions in the connections with
  quantum physics which have been unearthed.

  There is even more to challenge us in 4 dimensions emerging from the
  great discoveries of Donaldson. The role of Topology and the links
  with physics have yet to be fully explored and understood.

  I will attempt to provide a general survey putting Perelman's work
  in perspective and focusing on the future as well as the past.