Mathematical Sciences: Geometry of Yang-Mills Moduli Spaces

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The principal investigator will create a canonical metric for moduli spaces arising in four-dimensional Yang-Mills theory. He will extend the Donaldson program to spaces of 1-instantons over the four sphere and complex projective two-space. Particular attention will be paid to the nature of the boundary of moduli spaces; what is the nature of their strata and how do these strata fit together? The techniques used to approach these questions will be extensions of those used to understand simpler positive-definite manifolds. Four dimensional hypersurfaces have only recently become understandable to geometric topologists. Surprisingly, Donaldson used techniques from the theory of geometric partial differential equations to demonstrate the nonuniqueness of four dimensional space time. The principal investigator will extend this theory to more exotic hypersurfaces in a natural way. Much of this work has considerable impact on the superstring theories of gauge field physics.