We will first introduce the geography problem which is the existence problem for symplectic 4–manifolds with certain invariants. Then we will discuss the Cartwright–Steger surfaces, which are complex ball quotients (complex surfaces on the BMY line on the geography chart). Next, we will construct new simply connected, symplectic and exotic 4–manifolds with non-negative signatures that have the smallest Euler characteristics among the all known simply connected 4–manifolds with non-negative signatures. This is a joint work with A. Akhmedov and S.-K. Yeung.