Orbifolds, Higher Spin Curves, and Algebraic Structures

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DMS-0204824Takashi KimuraThe goal of this proposal is to investigate geometric constructions inspired by the analogy between the moduli space of curves together with higher spin structures and the moduli space of curves together with maps into orbifolds.The former is related to higher KdV integrable hierarchies while the latter is related to the quantum cohomology of orbifolds. We will analyze the resulting algebraic structures suggested by integrable systems and the topology of these spaces.The KdV equation is a nonlinear equation modeling the behavior of water waves in a straight channel and is the first of a hierarchy of equations whose solutions are mutually compatible. Such hierarchies and their generalizations are intimately related to surfaces endowed with additional fields possessinginternal symmetries. We will investigate this relationship and its ramifications.