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I will outline a conjectural relationship between logarithmic DT theory and GW theory.
This conjecture extends the usual GW/DT conjectures, and we now have a fairly complete
understanding of how these conjectures interact with degeneration techniques and
tropical geometry. I will then focus on perhaps the most basic instance of this setup,
namely the logarithmic enumerative geometry of the algebraic torus of dimension 3, and
explain what we know on the two sides. The theories have a number of interesting links
to nearby mathematics, for example with tropical refined curve counting, double
ramification cycles, and integrable hierarchies. I will try to explain these
connections.
The picture is based on recent and ongoing joint work with Maulik, but also touches upon work of Kennedy-Hunt, Shafi, and Urundolil Kumaran. |