Logarithmic enumerative geometry and moduli spaces

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Moduli spaces and enumerative geometry are central themes in algebraic geometry. Moduli spaces describe all geometric objects of a given type. Enumerative invariants are natural characteristic numbers of moduli spaces. They arise from studying the topology of these spaces and are often modeled on urve counting problems: counts of curves in a space subject to geometric constraints. The proposal aims to bring forward a fully-fledged theory of logarithmic enumerative geometry. Logarithmic geometry enriches objects of algebraic geometry with combinatorial data. It connects to traditional algebraic geometry by a process called degeneration. Logarithmic structures have been studied for decades but only recently have methods been developed to understand their combinatorial complexity. This is part of the subject of tropical geometry.The proposed advances in this logarithmic direction are guided by the interactions between the basic objects of enumerative geometry: the moduli space of curves, the space of stable maps, and the Hilbert scheme of embedded curves. The interactions have led to remarkable insights, such as the Gromov-Witten (GW)/Donaldson-Thomas (DT) correspondence, the cohomological field theory structure of GW theory, and numerous beautiful calculations on the moduli space of curves. We propose a new logarithmic GW/DT conjecture, a study of the algebraic structure of logarithmic GW theory using orbifolds, and a new logarithmic intersection theory on the moduli space of curves. These will lead to results in traditional enumerative geometry, such as a proof of the standard GW/DT correspondence for a very large class of Calabi-Yau threefolds. It will also build connections between different themes, such as mirror symmetry and the Hilbert schemes of points on a surface. Finally, it will open an entirely new direction in the study of the moduli space of curves using tropical geometry.