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I will survey various results on applications and extensions of Hofer-Wysocki-Zehnder's polyfold theory: - fiber products of polyfold Fredholm sections - equivariant transversality - existence and obstructions - equivariant fundamental class -

We develop a continuous version of Kasteleyn theory to show that partition functions for bead configurations on T_n may be expressed in terms of Fredholm determinants of certain operators on

Consider a space XX, such as a compact space of JJ-holomorphic stable maps with closed domain, that is the zero set of a Fredholm operator.