The Monopole (Bogomolnyi) equations are Geometric PDE in 3 dimensions that admit generalizations to Higher dimensional manifolds with special structures on them. Calabi Yau and G_2 manifolds are the main candidates for interesting solutions to these equations. There are conjectural relations between the submanifold theory of such manifolds and solutions to these PDE. These include theories counting calibrated cycles and defining invariants of Calabi Yau and G_2 structures. However, up to date there are no known solutions of the monopole equations. In this talk I plan to describe the monopole equations from scratch and also announce the first construction of solutions to these. |