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Explore the mathematical connections between orders and shuffle products on nestohedra in this conference talk that examines the evolution of Hopf algebras from the 1990s to recent developments. Learn about the foundational work on shuffles of permutations in Malvenuto-Reutenauer Hopf algebras and shuffles of rooted binary trees in Loday-Ronco Hopf algebras, which correspond to vertices of the permutohedron and associahedron respectively. Discover how the 1-skeletons of these polytopes relate to the weak Bruhat order and the Tamari lattice, and understand the "magic formula" established by Loday and Ronco in 2002 that links these orders to shuffle products. Follow the progression of research through extensions by Burgunder-Ronco and Loday-Ronco to faces of polytopes, and subsequent work by Palacios and Ronco on extending orders to all faces. Examine the contributions of Carr and Devadoss on graph associahedra orders, and Forcey and Ronco's extensions to polytope faces. Understand the speaker's recent collaborative work with Pierre-Louis Curien and Jovana Obradovic, which introduces conditions on families of nestohedra to obtain shuffle products (specifically tridendriform products) on faces, encompassing previous research while providing new examples. Gain insight into how this framework defines orders on nestohedra faces and establishes their connection to shuffle products, representing a significant advancement in the field of algebraic combinatorics and polytope theory.
