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Quotients of the multiplihedron as categorified associahedra
 Homotopy, Homology and Appl
, 2008
"... Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associah ..."
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Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of
GEOMETRIC REALIZATIONS OF THE MULTIPLIHEDRON AND ITS COMPLEXIFICATION.
"... Abstract. We realize Stasheff’s multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We also construct an algebraic variety that has the multiplihedron as its nonnegative real p ..."
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Abstract. We realize Stasheff’s multiplihedron geometrically as the moduli space of stable quilted disks. This generalizes the geometric realization of the associahedron as the moduli space of stable disks. We also construct an algebraic variety that has the multiplihedron as its nonnegative real part, and use it to define a notion of morphism of cohomological field theories. Contents
HOPF STRUCTURES ON THE MULTIPLIHEDRA
, 2009
"... Abstract. We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday–Ronco Hopf algebra. ..."
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Abstract. We investigate algebraic structures that can be placed on vertices of the multiplihedra, a family of polytopes originating in the study of higher categories and homotopy theory. Most compelling among these are two distinct structures of a Hopf module over the Loday–Ronco Hopf algebra.
Lifted generalized permutahedra and composition polynomials
"... Abstract. We introduce a “lifting ” construction for generalized permutohedra, which turns an ndimensional generalized permutahedron into an (n + 1)dimensional one. We prove that this construction gives rise to Stasheff’s multiplihedron from homotopy theory, and to the more general “nestomultiplih ..."
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Abstract. We introduce a “lifting ” construction for generalized permutohedra, which turns an ndimensional generalized permutahedron into an (n + 1)dimensional one. We prove that this construction gives rise to Stasheff’s multiplihedron from homotopy theory, and to the more general “nestomultiplihedra, ” answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this “composition polynomial ” arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal. Résumé. Nous introduisons une construction de “lifting ” (redressement) pour permutaèdres généralisés, qui transforme un permutaèdre généralisé de dimension n en un de dimension n + 1. Nous démontrons que cette construction conduit au multiplièdre de Stasheff à partir de la théorie d’homotopie, et aux “nestomultiplièdres, ” ce qui répond à deux questions de Devadoss et Forcey. Nous construisons une subdivision de n’importe quel permutaèdre généralisé dont les pièces sont indexées par compositions. La volume de chaque pièce est donnée par un polynôme dont nous recherchons les propriétés combinatoires. Nous montrons comment ce “polynôme de composition ” surgit naturellement dans l’interpolation d’une fonction exponentiel. Nous démontrons que ses coefficients sont strictement positifs, et nous conjecturons qu’ils sont unimodaux.
A combinatorial interpretation of the Catalan transform of the Catalan numbers
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Associahedral categories, particles and Morse functor
, 2009
"... Every smooth manifold contains particles which propagate. These form objects and ..."
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Every smooth manifold contains particles which propagate. These form objects and
Project description: Geometric combinatorial Hopf algebras and modules
"... In 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. The crucial mathematical ingredients included Hopf Algebras and subalgebras, in regard to the specific physical principle of renormalization in quantum electrodynamics and chromody ..."
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In 2008 several researchers made exciting advances in the application of algebra and combinatorics to particle physics. The crucial mathematical ingredients included Hopf Algebras and subalgebras, in regard to the specific physical principle of renormalization in quantum electrodynamics and chromodynamics. Renormalization refers to the addition of counterterms to a sequence of divergent integrals for probability
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"... This proposal is about combinatorial algebra, with a geometrical flavor. Together with students and collaborators I have been developing a diverse family of graded algebras and coalgebras, modules and comodules, often with Hopf and differential structures. Their common feature is that all are based ..."
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This proposal is about combinatorial algebra, with a geometrical flavor. Together with students and collaborators I have been developing a diverse family of graded algebras and coalgebras, modules and comodules, often with Hopf and differential structures. Their common feature is that all are based upon recursive sequences of convex polytopes. The key point of interest in each case is to see how the algebraic structure reflects the combinatorial structure, and vice versa. We are building upon the foundations laid by many other researchers, especially GianCarlo Rota, who most clearly saw the strength of this approach. The historical examples of Hopf algebras SSym and QSym, the MalvenutoReutenauer Hopf algebra and the quasisymmetric functions, can be defined using graded bases of permutations and boolean subsets respectively. Loday and Ronco used the fact that certain binary trees can represent both sorts of combinatorial objects to discover the Hopf algebra YSym lying between them. Chapoton capitalized on the fact that the three graded bases could actually be described as the vertex sets of polytope sequences, and defined larger algebras on the faces of the permutohedra, associahedra and cubes. The polytope sequences we study include those familiar examples as well as newer families such as the graph multiplihedra and composihedra. Simultaneously with our study of Hopf algebras we