ISS: Tim Perutz TODAY
Robert Lipshitz
As a reminder, Tim Perutz will speak on "Lagrangian correspondences for
fibered four-manifolds" TODAY in the Informal Symplectic Geometry Seminar,
at 5:30 p.m., in Math 507.
Tim's abstract:
The Heegaard Floer homology theory of Ozsvath and Szabo analyzes 3- and
4-manifolds in terms of their simplest pieces, namely, handles. I will
describe a related but still incomplete theory, which aims to analyze
4-manifolds via the simplest constituents of certain singular fibrations
over surfaces. The set-up involves, in three dimensions, circle-valued
Morse functions without extrema, and in four dimensions, the "singular
Lefschetz fibrations" introduced by Auroux, Donaldson and Katzarkov.
Like Heegaard Floer homology, the theory invokes symmetric products of
surfaces, and it appears to recover Seiberg-Witten gauge theory
symplectically. The crucial ingredient is a construction of Lagrangian
correspondences between symmetric products. The development of this theory
runs parallel to the study of a special class of submanifolds of
symplectic manifolds, the spherically-fibred coisotropic submanifolds.
According to work in progress, there are two interesting exact triangles
in symplectic Floer homology associated with these submanifolds -
counterparts of surgery and connected sum triangles in geometric topology.
In the first lecture, after a brief overview, I'll discuss "symplectic
Morse-Bott fibrations", their vanishing cycles, and associated moduli
spaces of pseudo-holomorphic curves.