Gauge theory on 2,3 and 4-dimensional manifolds and geometry

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Using monopole equations by Witten, we studied and obtained some results on 4-dimensional symplectic geometry. We studied the fundamental group and numerical invariants of a symplectic 4-manifold with posive c_1 (TX). As an application, we proved easily that the underlying 4-manifold of any rational surface can not be diffeomorphic to one of minimal surfaces of general type. As a special case, this theorem contains Hirzebruch's conjecture that there does not exist a minimal surface of general type which is diffeomorphic to S^2 * S^2. Furthermore, we proved that if (X,omega) is a symplectic 4-manifold such that c_1 (TX) [omega] [X] > 0 or X admits a metric of positive scalar curvature, then X must be diffeomorphic to a rational or ruled surface up to blow up down.Let GAMMA be a finite subgroup of SU (2). We have a specific fillable contact structure xi_0 on S^3/GAMMA induced from the standard contact structure on S^3. When we consider S^3/GAMMA with the fixed xi_0, we proved that the intersection form of any symplectically filling 4-manifold of (S^3/GAMMA, xi_0) is negative definite. For the proof, we showed a vanishing theorem on the Seiberg-Witten invariants. In the case GAMMA=E_8, we proved the intersection form of any minimal symplectically filling 4-manifold is equivalent to E_8. These are joint works with K.Ono.