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  • Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees

     Gálvez Carrillo, Maria Immaculada; Kock, Joachim; Tonks, Andrew
    Advances in mathematics
    Date of publication: 2014-03-20
    Journal article

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    We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. (C) 2013 Elsevier Inc. All rights reserved.

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    Thomason cohomology of categories  Open access

     Gálvez Carrillo, Maria Immaculada; Neumann, Frank; Tonks, Andrew
    Journal of pure and applied algebra
    Date of publication: 2013
    Journal article

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    We investigate cohomology and homology theories of categories with general coefficients given by functors on simplex categories first studied by Thomason. These generalize Baues¿Wirsching cohomology and homology of a small category, and coincide with Gabriel¿Zisman cohomology and homology of the simplicial nerve of the category. Thus Baues¿Wirsching cohomology of categories is seen to be a special case of simplicial cohomology. We analyze naturality and functoriality properties of these theories and construct associated spectral sequences for functors between small categories.

    We investigate cohomology and homology theories of categories with general coefficients given by functors on simplex categories first studied by Thomason. These generalize Baues–Wirsching cohomology and homology of a small category, and coincide with Gabriel–Zisman cohomology and homology of the simplicial nerve of the category. Thus Baues–Wirsching cohomology of categories is seen to be a special case of simplicial cohomology. We analyze naturality and functoriality properties of these theories and construct associated spectral sequences for functors between small categories.

    Postprint (author’s final draft)

  • GEOMATRIA ALGEBRAICA, SIMPLECTICA, ARITMETICA Y APLICACIONES

     Alberich Carramiñana, Maria; Amoros Torrent, Jaume; Barja Yañez, Miguel Angel; Elgueta Montó, Josep; Fernández Sánchez, Jesús; Ventura González Alonso, Daniel; Barbieri Solha, Romero; Miranda Galceran, Eva; Pascual Gainza, Pedro; Roig Marti, Agustin; Roig Maranges, Abdo; Feliu Trijueque, Elsienda; Gálvez Carrillo, Maria Immaculada; Casanellas Rius, Marta
    Participation in a competitive project

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  • Estructuras A-infinito en la opérada de cactus

     Gálvez Carrillo, Maria Immaculada; Lombardi, Leandro; Tonks, Andrew
    Encuentro de Topología
    Presentation's date: 2012-10-19
    Presentation of work at congresses

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    Diversas versiones de la opérada de cactus inicialmente definida por Voronov han sido estudiadas. Es conocida su equivalencia débil con la opérada de pequeños discos. Se conoce pues que la opérada de cactus admite una acción de la opérada de Gerstenhaber salvo homotopía. En este proyecto, nuestro objetivo es obtener una realización explícita de dicha acción. Por el momento, hemos construido una acción explícita de la opérada A8 en la opérada de cactus, que presentamos en este póster

  • André spectral sequences for Baues¿Wirsching cohomology of categories

     Gálvez Carrillo, Maria Immaculada; Neumann, Frank; Tonks, Andrew
    Journal of pure and applied algebra
    Date of publication: 2012-04-30
    Journal article

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  • Homotopy Batalin-Vilkovisky Algebras

     Gálvez Carrillo, Maria Immaculada; Tonks, Andrew; Vallette, Bruno
    Journal of noncommutative geometry
    Date of publication: 2012
    Journal article

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    Central cohomology operations and K-theory  Open access

     Gálvez Carrillo, Maria Immaculada; Whitehouse, Sarah
    Date: 2012-04
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    For stable degree zero operations, and also for additive unstable operations of bidegree (0,0), it is known that the centre of the ring of operations for complex cobordism is isomorphic to the corresponding ring of connective complex K-theory operations. Similarly, the centre of the ring of BP operations is the corresponding ring for the Adams summand of p-local connective complex K-theory. Here we show that, in the additive unstable context, this result holds with BP replaced by BP for any n. Thus, for all chromatic heights, the only central operations are those coming from K-theory

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    Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees  Open access

     Gálvez Carrillo, Maria Immaculada; Tonks, Andrew; Kock, Joachim
    Date: 2012-07
    Report

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    We prove a Faa di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs

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    Thomason cohomology of categories  Open access

     Gálvez Carrillo, Maria Immaculada; Neumann, Frank; Tonks, Andrew
    Date: 2012-08
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    We introduce cohomology and homology theories for small categories with general coefficient systems from simplex categories first studied by Thomason. These theories generalize at once Baues-Wirsching cohomology and homology and other more classical theories. We analyze naturality and functoriality properties of these theories and construct associated spectral sequences for functors between small categories.

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    André spectral sequences for Baues-Wirsching cohomology of categories  Open access

     Gálvez Carrillo, Maria Immaculada; Neumann, Frank; Tonks, Andrew
    Date: 2011-12-16
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    We construct spectral sequences in the framework of Baues-Wirsching cohomology and homology for functors between small categories and analyze particular cases including Grothendieck fibrations. We also give applications to more classical cohomology and homology theories including Hochschild-Mitchell cohomology and those studied before by Watts, Roos, Quillen and others

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    Homotopy Batalin-Vilkovisky algebras  Open access

     Gálvez Carrillo, Maria Immaculada; Tonks, Andrew; Vallette, Bruno
    Date: 2011-03-30
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    This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

  • Álgebras de vértices, opéradas y complejo de De Rham quiral

     Gálvez Carrillo, Maria Immaculada
    Date: 2011-12-16
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    Homotopy Gerstenhaber structures and vertex algebras  Open access

     Gálvez Carrillo, Maria Immaculada; Gorbounov, V.; Tonks, Andrew
    Applied categorical structures
    Date of publication: 2010
    Journal article

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    We provide a simple construction of a G∞-algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie∞ structure related to the bracket of Courant

    Postprint (author’s final draft)

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    Infinite sums of Adams operations and cobordism.  Open access

     Gálvez Carrillo, Maria Immaculada; Whitehouse, Sarah
    Mathematische Zeitschrift
    Date of publication: 2005
    Journal article

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    The elements of various algebras of stable degree zero operations in p-local K-theory can be described explicitly as certain infinite sums of Adams operations [11, 9]. Here we show how to make sense of the same expressions for MU(p) and BP, thus identifying the “Adams subalgebra” of the algebras of operations. We prove that the Adams subalgebra is the centre of the ring of degree zero operations

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    Differential operators and the Witten genus for projective spaces and Milnor manifolds  Open access

     Gálvez Carrillo, Maria Immaculada; Tonks, Andrew
    Mathematical proceedings of the Cambridge Philosophical Society
    Date of publication: 2003
    Journal article

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    A $genus$ (in the sense of Hirzebruch [4]) is a multiplicative invariant of cobordism classes of manifolds. Classical examples include the numerical invariants given by the signature and the $\widehat{A}$- and Todd genera. More recently genera were introduced which take as values modular forms on the upper half-plane, $\frak{h}=\{\,\tau\;|\;\mathrm{Im}(\tau)>0\,\}$. The main examples are the elliptic genus $\phi_{ell}$ and the Witten genus $\phi_W$; we refer the reader to the texts [7] or [9] for details

    Postprint (author’s final draft)