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Explore the intersection of tropical geometry and topological field theory in this differential geometry and physics seminar lecture. Discover how tropical geometry creates a bridge between complex and combinatorial worlds, enabling the computation of curve-counting invariants through a piecewise-linear "tropical" limit. Learn about Mikhalkin's groundbreaking insight that Gromov-Witten invariants can be recovered from tropical curves, examined through the lens of topological field theory and functional integration. Understand the topological sigma model and how its localization equations naturally admit tropicalization, allowing reproduction of Gromov-Witten invariants using standard cohomological BRST methods without reformulating the functional integral in terms of the tropical semifield. Examine how the tropical limit of topological sigma models leads to geometries that no longer require complex structures but are instead based on nilpotent structures on singular foliated manifolds. Gain insights into recent developments connecting the tropological sigma model to Hořava's topological quantum gravity for Ricci Flow, as discussed in collaborative work with Emil Albrychiewicz and Viola Zixin Zhao.
