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Explore the combinatorial theory of permutation flows and their applications to flow polytopes in this mathematical lecture. Delve into the Danilov-Karzanov-Koshevoy (DKK) method for obtaining regular unimodular triangulations of flow polytopes on acyclic directed graphs with unique source and sink vertices and unit netflow. Examine the conjecture by González D'León et al. regarding the lattice structure of dual graphs in DKK triangulations, along with recent independent proofs by Bell and Ceballos, and by Berggren and Serhiyenko. Learn about a novel combinatorial approach to studying these lattices through permutation flows, including how to derive formulas for h*-polynomials of flow polytopes (G-Eulerian polynomials). Discover the extension of DKK triangulations to flow polytopes with nonnegative integer netflows and understand a new proof of the generalized Lidskii formula for calculating flow polytope volumes. This presentation was recorded during the thematic meeting "Beyond Permutahedra and Associahedra" at the Centre International de Rencontres Mathématiques in Marseille, France.
