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.    Classical knot theory deals with classifying knotted one-dimensional curves in three-dimensional space.  My proposed research will study knotted two-dimensional surfaces in symplectic four-manifolds.  Note that dimensions of both the knotted object and the ambient space containing it have gone up by one, and consequently the classification of such knotted two-dimensional surfaces turns out to be a rather daunting task.  My research will focus on the subclass of two-dimensional surfaces that inherit a symplectic substructure from the ambient symplectic four-manifold containing them.  Experts conjecture that the number and the variety of knotted two-dimensional surfaces within this subclass measure the topological complexity of the ambient symplectic four-manifold containing those knotted surfaces.  My previous research has shown some evidence of this conjecture and I hope that my future work in this direction can contribute towards the classification of symplectic four-manifolds.