Ho, Chung-I2011-08-162011-08-162011-07https://hdl.handle.net/11299/113122University of Minnesota Ph.D. dissertation. July 2011. Major: Mathematics. Advisor: Tian-Jun Li. 1 computer file (PDF); v, 93 pages.In the first part, we study existence of the Lefschetz decomposition for de Rham cohomology, which is characterized by the strong Lefschetz property. A new spectral sequence for symplectic manifolds is also defined. In the second part, we show that the Lagrangian Luttinger surgery preserves the Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.en-USMathematicsThesis or Dissertation