The proposal concerns several aspects of four dimensional topology and geometry. Some of the projects on the topological side are: defining smooth and symplectic invariants and studying their properties, classifying symplectic Calabi-Yau 4-manifolds and classifying Lagrangian and symplectic surfaces in rational and ruled manifolds, investigating smooth and symplectic mapping class groups. And on the geometric side, the PI is comparing tamed, compatible and integrable almost complex structures on 4-manifolds.
An n dimensional manifold is a large scale space that locally looks like the Euclidean space of dimension n. In particular, the space-time universe we live in is a four dimensional manifold. On an even dimensional manifold, a symplectic structure is a geometric structure that underlies some fundamental equations of classical and quantum physics. Thus symplectic four manifolds play a central role in both mathematics and physics. This research project aims to gain some new understandings of the general shape of symplectic four manifolds, using a variety of techniques from topology, algebraic geometry and differential geometry.
