@article{GuoPaychaZhang2017,
  author    = {Guo, Li and Paycha, Sylvie and Zhang, Bin},
  title     = {Algebraic Birkhoff factorization and the Euler-Maclaurin formula on cones},
  series = {Duke mathematical journal},
  volume    = {166},
  journal   = {Duke mathematical journal},
  number    = {3},
  publisher = {Duke Univ. Press},
  address   = {Durham},
  issn      = {0012-7094},
  doi       = {10.1215/00127094-3715303},
  pages     = {537 -- 571},
  year      = {2017},
  abstract  = {We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler-Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler-Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.},
  language  = {en}
}
@article{ClavierGuoPaychaetal.2019,
  author    = {Clavier, Pierre J. and Guo, Li and Paycha, Sylvie and Zhang, Bin},
  title     = {An algebraic formulation of the locality principle in renormalisation},
  series = {European Journal of Mathematics},
  volume    = {5},
  journal   = {European Journal of Mathematics},
  number    = {2},
  publisher = {Springer},
  address   = {Cham},
  issn      = {2199-675X},
  doi       = {10.1007/s40879-018-0255-8},
  pages     = {356 -- 394},
  year      = {2019},
  abstract  = {We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs.},
  language  = {en}
}
@article{GuoPaychaZhang2014,
  author    = {Guo, Li and Paycha, Sylvie and Zhang, Bin},
  title     = {Conical zeta values and their double subdivision relations},
  series = {Advances in mathematics},
  volume    = {252},
  journal   = {Advances in mathematics},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0001-8708},
  doi       = {10.1016/j.aim.2013.10.022},
  pages     = {343 -- 381},
  year      = {2014},
  abstract  = {We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.},
  language  = {en}
}
@article{ClavierGuoPaychaetal.2020,
  author    = {Clavier, Pierre and Guo, Li and Paycha, Sylvie and Zhang, Bin},
  title     = {Locality and renormalization: universal properties and integrals on trees},
  series = {Journal of mathematical physics},
  volume    = {61},
  journal   = {Journal of mathematical physics},
  number    = {2},
  publisher = {American Institute of Physics},
  address   = {College Park, Md.},
  issn      = {0022-2488},
  doi       = {10.1063/1.5116381},
  pages     = {19},
  year      = {2020},
  abstract  = {The purpose of this paper is to build an algebraic framework suited to regularize branched structures emanating from rooted forests and which encodes the locality principle. This is achieved by means of the universal properties in the locality framework of properly decorated rooted forests. These universal properties are then applied to derive the multivariate regularization of integrals indexed by rooted forests. We study their renormalization, along the lines of Kreimer's toy model for Feynman integrals.},
  language  = {en}
}