Overview of Some Appearances of Multivalued Solutions in Differential Geometry

An interesting development, over the past decade or so, has been the appearance of “multivalued” objects (such as functions, forms, spinors) in various branches of differential geometry. These include gauge theory, calibrated geometry and adiabatic limits of structures on manifolds. The developments extend more classical constructions in the case of Riemann surfaces. In the first part of the talk we review, in outline, some of the geometric and analytic aspects of these developments (related to other talks in this workshop). In the last part we focus in more detail on the adiabatic description of co-associative fibred G2 manifolds and research directions which these suggest.

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