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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

15 votes
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The core question of topology

As others have noted, it's hopeless to try to answer this question for general topological spaces. However, there are a few positive results if you assume, say, that X and Y are both simply connected …
Mike Usher's user avatar
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4 votes
Accepted

Legendrian homotopy of curves in a contact structure?

In general, the (parametric) h-principle for Legendrian immersions implies that Legendrian immersions f:L->(M,\xi) are classified up to homotopy (through Legendrian immersions) by the following bundle …
Mike Usher's user avatar
  • 2,927
13 votes

The "miracle" of Heegard Floer.

From Szabó's delightfully understated response (pdf) to receiving the Veblen prize: The joint work with Peter Ozsváth which is noted here grew out of our attempts to understand Seiberg–Witten moduli …
Mike Usher's user avatar
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