In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int
\sigma_2\ge 0$, then topologically $M$ is a quotient of sphere.

One of the fundamental problems in the classification of complex surfaces is to find a new family of simply connected surfaces with p_g = 0 and K^2 > 0. In this
talk, I will sketch how to construct a new family of simply connected symplectic 4- manifolds using a rational blow-down surgery and how to show that such 4-manifolds
admit a complex structure using a Q-Gorenstein smoothing theory. In particular, I will show explicitly how to construct a simply connected minimal surface of general
type with p_g = 0 and K^2 = 3.

If time allows, I will also sketch how to construct a simply
connected, minimal, symplectic 4-manifold with b_+2 = 1 (equivalently, p_g = 0) and K^2 = 4 using a rational blow-down surgery.

After discussing the basics of contact surgery in dimension 3, and introducing contact Ozsvath-Szabo invarinats, we show that a Seifert fibered 3-manifold does admit a positive tight contact structure unless it is orientation preserving diffeomorphic to the result of (2n-1)-surgery along the T(2,2n+1) torus knot (for some positive integer n).