@article{MickelssonPaycha2010,
  author    = {Mickelsson, Jouko and Paycha, Sylvie},
  title     = {The logarithmic residue density of a generalized Laplacian},
  series = {Journal of the Australian Mathematical Society},
  volume    = {90},
  journal   = {Journal of the Australian Mathematical Society},
  number    = {1},
  publisher = {Cambridge Univ. Press},
  address   = {Cambridge},
  issn      = {0263-6115},
  doi       = {10.1017/S144678871100108X},
  pages     = {53 -- 80},
  year      = {2010},
  abstract  = {We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold definesan invariant polynomial-valued differential form. We express it in terms of a finite sum of residues ofclassical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas providea pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in four dimensions and for atwisted Dirac operator on a flat space of any dimension. These correspond to special cases of a moregeneral formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either aCampbell-Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.},
  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}
}
@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{LevyJimenezPaycha2016,
  author    = {Levy, Cyril and Jimenez, Carolina Neira and Paycha, Sylvie},
  title     = {THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS},
  series = {Transactions of the American Mathematical Society},
  volume    = {368},
  journal   = {Transactions of the American Mathematical Society},
  publisher = {American Mathematical Soc.},
  address   = {Providence},
  issn      = {0002-9947},
  doi       = {10.1090/tran/6369},
  pages     = {1051 -- 1095},
  year      = {2016},
  abstract  = {Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) noninteger order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace-class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the \$ \zeta \$-function at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.},
  language  = {en}
}
@article{BellingeriFrizPaychaetal.2022,
  author    = {Bellingeri, Carlo and Friz, Peter and Paycha, Sylvie and Preiß, Rosa Lili Dora},
  title     = {Smooth rough paths, their geometry and algebraic renormalization},
  series = {Vietnam journal of mathematics},
  volume    = {50},
  journal   = {Vietnam journal of mathematics},
  number    = {3},
  publisher = {Springer},
  address   = {Singapore},
  issn      = {2305-221X},
  doi       = {10.1007/s10013-022-00570-7},
  pages     = {719 -- 761},
  year      = {2022},
  abstract  = {We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the key to a purely algebraic form of Lyons' extension theorem, the renormalization of rough paths following up on [Bruned et al.: A rough path perspective on renormalization, J. Funct. Anal. 277(11), 2019], as well as a related notion of "sum of rough paths". We first develop our ideas in a geometric rough path setting, as this best resonates with recent works on signature varieties, as well as with the renormalization of geometric rough paths. We then explore extensions to the quasi-geometric and the more general Hopf algebraic setting.},
  language  = {en}
}