Geometry of vector bundles and moduli spaces of instantons

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This research program is concerned with understanding the topology and geometry of various moduli spaces of vector bundles equipped with special structures. Such a program lies in the mathematical field of differential geometry, the study of calculus and differential equations on manifolds, and more specifically it lies in the field of gauge theory, the study of vector bundles and connections.****There have been ongoing efforts in the mathematics and physics worlds to classify Yang-Mills instantons, which are connections whose curvature minimizes a certain energy. The focus of this research program is to do this classification on certain non-compact manifolds, that is to gain knowledge about the geometry and topology of the spaces of instantons living on those spaces, modulo natural equivalences. Ideally, one could express this space, a space of solutions to a difficult partial differential equation, in terms of solutions to more tractable equations (ordinary differential equation, algebraic equation). One then describes the geometry of the starting space through such a lens. Working on a non-compact base manifold creates serious difficulties and tools must be developed to address them.****Overcoming these difficulties and providing information about these moduli spaces allows a better understanding of the physics of instantons, provides valuable information about the underlying manifolds, and is interesting math on its own sake.**********************