Exotic manifolds are smooth manifolds that are homeomorphic but not diffeomorphic to each other. Two major open problems in low dimensional topology are whether there exist exotic CP^2 and S^2 x S^2. In literature there are many constructions of simply connected, minimal, symplectic 4-manifolds with more than one smooth structures. Here two interesting questions are whether one can give different constructions of such manifolds, and how small manifolds (i.e. manifolds with small Euler characteristics) one can obtain. In this talk, first I will construct exotic CP^2 # 6(-CP^2) and CP^2 # 7(-CP^2) by using Namikawa-Ueno’s classification of singular fibers in pencils of genus two curves. This is a joint work with A. Akhmedov. Next, I will discuss our constructions of exotic copies of n(S^2xS^2), the connected sum of n copies of S^2xS^2, for every odd integer n>= 27. This is a joint work with A. Akhmedov and D. Park.

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