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Explore a mathematical seminar presentation examining the topological bounds of tropical varieties, focusing on groundbreaking developments since Mikhalkin's construction of a non-planar tropical cubic curve in R^3 of genus 1. Learn about recent discoveries showing how the upper bound on the top Betti number of combinatorial tropical varieties depends on codimension - a distinct contrast from complex cases. Understand how for any three positive integers d, m, and k, tropical varieties of degree d, dimension m, and codimension k can be constructed with top Betti numbers equal to k times the upper bound of complex varieties' top Betti numbers. Delve into the Floor composition method used for these constructions, which proves maximal for lower dimensions and degrees. This collaborative research presentation by Lucia Lopez de Medrano, conducted with Benoît Bertrand and Erwan Brugallé, advances our understanding of non-realizable tropical varieties and their topological nature.
