Donaldson-Thomas theory and tropical geometry

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The project is in algebraic geometry and broadly speaking, lies at the intersection of tropical geometry and enumerative geometry. Enumerative geometry is a subject of classical and contemporary interest, and its core goals are to probe the geometry of certain spaces (algebraic manifolds) by extracting numerical information associated to the set of geometric objects, e.g. bundles or submanifolds. The student will explore logarithmic Donaldson-Thomas (DT) theory, which is a young subject, and explore its relationship to tropical geometry, and to Gromov-Witten theory. The likely direction of the thesis will be a study of the newly constructed logarithmic Hilbert scheme of points and its basic topology, progressing towards a general study of log DT invariants in the case of surfaces.