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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
15
votes
Accepted
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 …
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 …
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 …