Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen OPUS4-49226 Wissenschaftlicher Artikel Clavier, Pierre J.; Guo, Li; Paycha, Sylvie; Zhang, Bin An algebraic formulation of the locality principle in renormalisation 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. Cham Springer 2019 39 European Journal of Mathematics 5 2 356 394 10.1007/s40879-018-0255-8 Institut für Mathematik OPUS4-44476 Teil eines Buches Clavier, Pierre J.; Guo, Li; Paycha, Sylvie; Zhang, Bin Renormalisation and locality Zürich European Mathematical Society Publishing House 2020 47 Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2 978-3-03719-205-4 print 85 132 10.4171/205 Institut für Mathematik OPUS4-55257 misc Paycha, Sylvie Interview with Pierre Cartier New York Springer 2017 7 The mathematical intelligencer 39 15 21 10.1007/s00283-016-9673-y Institut für Mathematik OPUS4-55331 Wissenschaftlicher Artikel Guo, Li; Paycha, Sylvie; Zhang, Bin Algebraic Birkhoff factorization and the Euler-Maclaurin formula on cones 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. Durham Duke Univ. Press 2017 35 Duke mathematical journal 166 3 537 571 10.1215/00127094-3715303 Institut für Mathematik OPUS4-54085 Wissenschaftlicher Artikel Paycha, Sylvie When the market wins over research and higher education In this chapter, an overview of systematic eradication of basic science foci in European universities in the last two decades is given. This happens under the slogan of optimisation of the university education to the needs and demands of the society. It is pointed out that reliance on "market demands" brings with it long-term deficiencies in the maintenance of basic and advanced knowledge construction in societies necessary for long-term future technological advances. University policies that claim improvement of higher education towards more immediate efficiency may end up with the opposite effect of affecting its quality and long term expected positive impact on society. Cham Springer 2018 6 Sustainable Futures for Higher Education : the Making of Knowledge Makers 7 978-3-319-96035-7 23 28 10.1007/978-3-319-96035-7_2 Institut für Mathematik OPUS4-58767 Wissenschaftlicher Artikel Azzali, Sara; Paycha, Sylvie Spectral zeta-invariants lifted to coverings The canonical trace and the Wodzicki residue on classical pseudo-differential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral zeta-invariants using lifted defect formulae which express discrepancies of zeta-regularised traces in terms of Wodzicki residues. We derive Atiyah's L-2-index theorem as an instance of the Z(2)-graded generalisation of the canonical lift of spectral zeta-invariants and we show that certain lifted spectral zeta-invariants for geometric operators are integrals of Pontryagin and Chern forms. Providence, RI American Mathematical Society 2020 42 Transactions of the American Mathematical Society 373 9 6185 6226 10.1090/tran/8067 Institut für Mathematik OPUS4-38050 Wissenschaftlicher Artikel Guo, Li; Paycha, Sylvie; Zhang, Bin Conical zeta values and their double subdivision relations 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. San Diego Elsevier 2014 39 Advances in mathematics 252 343 381 10.1016/j.aim.2013.10.022 Institut für Mathematik OPUS4-45671 Wissenschaftlicher Artikel Levy, Cyril; Jimenez, Carolina Neira; Paycha, Sylvie THE CANONICAL TRACE AND THE NONCOMMUTATIVE RESIDUE ON THE NONCOMMUTATIVE TORUS 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. Providence American Mathematical Soc. 2016 45 Transactions of the American Mathematical Society 368 1051 1095 10.1090/tran/6369 Institut für Mathematik OPUS4-42517 Wissenschaftlicher Artikel Mickelsson, Jouko; Paycha, Sylvie The logarithmic residue density of a generalized Laplacian 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. Cambridge Cambridge Univ. Press 2010 28 Journal of the Australian Mathematical Society 90 1 53 80 10.1017/S144678871100108X Mathematisch-Naturwissenschaftliche Fakultät OPUS4-61004 Wissenschaftlicher Artikel Clavier, Pierre; Guo, Li; Paycha, Sylvie; Zhang, Bin Locality and renormalization: universal properties and integrals on trees 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. College Park, Md. American Institute of Physics 2020 19 Journal of mathematical physics 61 2 10.1063/1.5116381 Institut für Mathematik OPUS4-41368 misc Mickelsson, Jouko; Paycha, Sylvie The logarithmic residue density of a generalized Laplacian We show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah-Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell-Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula. 2011 28 Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe 649 urn:nbn:de:kobv:517-opus4-413680 10.25932/publishup-41368 Mathematisch-Naturwissenschaftliche Fakultät OPUS4-62393 Wissenschaftlicher Artikel Bellingeri, Carlo; Friz, Peter; Paycha, Sylvie; Preiß, Rosa Lili Dora Smooth rough paths, their geometry and algebraic renormalization 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. Singapore Springer 2022 43 Vietnam journal of mathematics 50 3 719 761 10.1007/s10013-022-00570-7 Institut für Mathematik