In this talk I will describe the progress that has been made so far
concerning the existence of global strong solutions to the
L^{2}-critical defocusing semilinear Schr\"odinger equation. A long
standing conjecture in the area is the existence
of a unique global strong L^{2}
solution to the equation that in addition scatters to a free solution as
time goes to infinity. I will demonstrate the proofs of partial results
towards an attempt for a final resolution of this conjecture.
I will concentrate on the low dimensions but give the flavor of the results in higher dimensions for general or spherically symmetric initial data in certain Sobolev spaces.
Many authors have contributed to the theory of this equation. I will convey my personal involment to the problem and the results that I have obtained recently. Part of my work is in collaboration with D. De Silva, N. Pavlovic, G.
Staffilani, J. Colliander and M. Grillakis.

Generalized equations of motion for the Weber-Clebsch potentials that
reproduce Navier-Stokes dynamics are derived. These depend on a new
parameter, with the dimension of time, and reduce to the Ohkitani and
Constantin equations in the singular special case where the new
parameter vanishes.
Let us recall that Ohkitani and Constantin found that the diffusive
Lagrangian map became noninvertible under time evolution and required
resetting for its calculation. They proposed that high frequency of
resetting was a diagnostic for vortex reconnection.
Direct numerical simulations of the generalized equations of motion are
performed. The Navier-Stokes dynamics is well reproduced at small enough
Reynolds number without resetting. Computation at higher Reynolds
numbers is achieved by performing resettings. The interval between
successive resettings is found to abruptly increase when the new
parameter is varied from zero to a value much smaller than the resetting
interval.