The $4$-genus of a knot is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 
 $4$-genus and related invariants of homology classes is the \emph{Thom Conjecture}, proved by Kronheimer-Mrowka, and its symplectic extension 
 due to Ozsvath-Szabo, which say that \textit{closed} symplectic surfaces minimize genus. Suppose $(X, \omega) $ is a symplectic 4-manifold with 
 contact type bounday $\partial X $ and $\Sigma $ is a symplectic surface in $X$ such that $\partial \Sigma $ is a transverse knot in $\partial 
 X $. In this talk we show that there is a closed symplectic 4-manifold $Y $ with a closed symplectic submanifold $ S$ such that the pair $(X, 
 \Sigma) $ embeds symplectically into $(Y, S) $.  This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 
 4-genus of fibered knots in $\mathbb{S}^3$. We also discuss the symplectic convexity of unit circle bundle in a Hermitian holomorphic line bundle 
 over a Riemann surface. This is joint work with Prof. Siddhartha Gadgil.