U(1)-invariant special Lagrangian 3-folds in C^3 and special Lagrangian   fibrations
  This is a survey of the author's series of three papers math.DG/0111324, math.DG/0111326, math.DG/0204343 using analysis to investigate special Lagrangian 3-folds (SL 3-folds) in C^3 invariant under the U(1)-action (z_1,z_2,z_3) --> (gz_1,g^{-1}z_2,z_3) for unit complex numbers g, and their sequel math.DG/0011179 on special Lagrangian fibrations and the SYZ Conjecture.   We briefly present the main results of these four long papers, giving some explanation and motivation, but no proofs. The aim is to make the results and ideas accessible to String Theorists and others who have an interest in special Lagrangian 3-folds and fibrations, but have no desire to read pages of technical analysis.   Let N be an SL 3-fold in C^3 invariant under the U(1)-action above. Then |z_1|^2-|z_2|^2=2a on N for some real number a. Locally, N can be written as a kind of graph of functions u,v : R^2 --> R satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a=0 the equations may have singular points where u,v are not differentiable, which leads to analytic difficulties.   We prove existence and uniqueness results for solutions u,v on domains S in R^2 with boundary conditions, including singular solutions. We study their singularities, giving a rough classification by multiplicity and type. We prove the existence of large families of fibrations of open subsets of C^3 by U(1)-invariant SL 3-folds, including singular fibres. Finally, we use these fibrations as local models to draw conclusions about the SYZ Conjecture on Mirror Symmetry of Calabi-Yau 3-folds. 
Dominic Joyce