Abstract: |
Special Lagrangian submanifolds are area-minimizing Lagrangian submanifolds of Calabi--Yau manifolds, discovered by Harvey and Lawson in 1980's as a special class of area-minimizing submanifolds. It's naturally of, interest to differential geometers, but it's also been important in string theory more recently; for instance an interesting conjecture was posed by Strominger, Yau and Zaslow, called the SYZ conjecture. It seems very difficult however even to find a mathematically-correct formulation of the SYZ conjecture, which requires us to have a deep understanding of singularities of special Lagrangian submanifolds. Another interesting problem is to define counting invariants of special Lagrangian submanifolds which may be related to Donaldson--Thomas invariants by mirror symmetry. One can use algebraic geometry for the definition of Donaldson--Thomas invariants, but cannot use it directly for special Lagrangians. There're two examples of "counting" in differential geometry: one is Donaldson invariants which count Yang--Mills anti-self-dual instantons in dimension 4, and the other is Gromov--Witten invariants which count pseudo-holomorphic curves in symplectic manifolds. Both instantons in dimension 4 and pseudo-holomorphic curves have only isolated singularities but special Lagrangians may have non-isolated singularities. It's difficult even to study isolated singularities of special Lagrangian submanifolds, but I've developed a fairly nice theory on "simple" isolated singularities, using geometric measure theory and Lagrangian Floer theory. I'll give more details in the seminar. |