Topology of symplectic four-manifolds

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Topology is a branch of geometry where two geometric objects are considered to be equal if they can be continuously deformed from one to the other. Four-manifolds are four-dimensional geometric objects that do not have any sharp corner or an edge. They are of fundamental importance as our universe itself can be modeled using them. Most elementary examples of four-manifolds are the complex surfaces, which are defined by systems of equations in complex numbers. It is now known that the vast majority of four-manifolds are not complex surfaces. My proposed research will study a more general class of four-manifolds, called symplectic four-manifolds. Although most four-manifolds are not symplectic either, experts conjecture that all four-manifolds can be built up from symplectic four-manifolds via some cut-and-paste operations. The complete classification of symplectic four-manifolds is still far out of our reach with our current knowledge, but topologists have been making significant progress in recent years.