Explorations in Symplectic Geometry: Hofer-Zehnder Conjecture and Lagrangian Tori in $S^2\times S^2$

This dissertation is about two different aspects of symplectic geometry. The first part is about the Hofer-Zehnder conjecture. Arnold conjecture says that the number of $1$-periodic orbits of a Hamiltonian diffeomorphism is greater than or equal to the dimension of the Hamiltonian Floer homology. In 1994, Hofer and Zehnder conjectured that there are infinitely many periodic orbits if the equality does not hold. We showed that the Hofer-Zehnder conjecture is true for semipositive symplectic manifolds with semisimple quantum homology. The second part is about Lagrangian submanifolds. In [Fuk+12], K. Fukaya, Y. Oh, H. Ohta, and K. Ono (FOOO) obtained the monotone symplectic manifold $S^2\times S^2$ by resolving the singularity of a toric degeneration of a Hirzebruch surface. They identified a continuum of toric fibers in the resolved toric degeneration that are not Hamiltonian isotopic to the toric fibers of the standard toric structure on $S^2\times S^2$. We provided a comprehensive classification: for any toric fiber in FOOO's construction of $S^2\times S^2$, we determined whether it is Hamiltonian isotopic to a toric fiber of the standard toric structure of $S^2\times S^2$.