John Douglas Moore studies differential geometry and geometric analysis. His UCSB research page describes work on minimal surfaces and related variational problems.[ref]
His Lecture Notes on Seiberg-Witten Invariants by John Douglas Moore develop gauge-theoretic methods for four-manifolds. The treatment of Hodge theory connects the curvature of a line-bundle connection with its harmonic representative: analytic equations select distinguished representatives of topological classes. This provides a bridge from differential equations to invariants of smooth manifolds.[ref]
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