42517
2010
2011
eng
53
80
28
1
90
article
Cambridge Univ. Press
Cambridge
1
2011-05-18
2011-02-01
--
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.
Journal of the Australian Mathematical Society
10.1017/S144678871100108X
0263-6115
1446-8107
<a href="http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-413680">Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 649</a>
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Jouko Mickelsson
Sylvie Paycha
eng
uncontrolled
residue
eng
uncontrolled
index
eng
uncontrolled
Dirac operators
Mathematik
Mathematisch-Naturwissenschaftliche Fakultät
Referiert
41368
2011
2019
eng
28
649
postprint
1
2019-02-25
2019-02-25
--
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.
Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
10.25932/publishup-41368
urn:nbn:de:kobv:517-opus4-413680
1866-8372
online registration
Journal of the Australian Mathematical Society 90 (2011), pp. 53–80 DOI 10.1017/S144678871100108X
<a href="http://publishup.uni-potsdam.de/opus4-ubp/frontdoor/index/index/docId/42517">Bibliographieeintrag der Originalveröffentlichung/Quelle</a>
Keine öffentliche Lizenz: Unter Urheberrechtsschutz
Jouko Mickelsson
Sylvie Paycha
Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
649
eng
uncontrolled
residue
eng
uncontrolled
index
eng
uncontrolled
Dirac operators
Mathematik
open_access
Mathematisch-Naturwissenschaftliche Fakultät
Referiert
Open Access
Cambridge University Press (CUP)
Universität Potsdam
https://publishup.uni-potsdam.de/files/41368/pmnr649.pdf
38050
2014
2014
eng
343
381
39
252
article
Elsevier
San Diego
1
--
--
--
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.
Advances in mathematics
10.1016/j.aim.2013.10.022
0001-8708
1090-2082
wos:2014
WOS:000330153100012
Zhang, B (reprint author), Sichuan Univ, Yangtze Ctr Math, Chengdu 610064, Peoples R China., liguo@rutgers.edu; paycha@math.uni-potsdam.de; zhangbin@scu.edu.cn
NSF [DMS 1001855]; NSFC [11071176, 11221101]
Li Guo
Sylvie Paycha
Bin Zhang
eng
uncontrolled
Convex cones
eng
uncontrolled
Conical zeta values
eng
uncontrolled
Smooth cones
eng
uncontrolled
Decorated cones
eng
uncontrolled
Subdivisions
eng
uncontrolled
Multiple zeta values
eng
uncontrolled
Shuffles
eng
uncontrolled
Quasi-shuffles
eng
uncontrolled
Fractions with linear poles
eng
uncontrolled
Shintani zeta values
Institut für Mathematik
Referiert
45671
2016
2016
eng
1051
1095
45
368
article
American Mathematical Soc.
Providence
1
--
--
--
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.
Transactions of the American Mathematical Society
10.1090/tran/6369
0002-9947
1088-6850
wos2016:2019
WOS:000366330100010
Levy, C (reprint author), Ctr Univ Jean Francois Champollion, Pl Verdun, F-81000 Albi, France., levy@math.uni-potsdam.de; cyril.levy@univ-jfc.fr; paycha@math.uni-potsdam.de
importub
2020-03-22T20:29:01+00:00
filename=package.tar
73f596a307ee4360636407e08d878f50
Cyril Levy
Carolina Neira Jimenez
Sylvie Paycha
Institut für Mathematik
Referiert
Import
55257
2017
2017
eng
15
21
7
39
other
Springer
New York
1
2016-11-02
2016-11-02
--
Interview with Pierre Cartier
The mathematical intelligencer
Potsdam, December 4th, 2014
10.1007/s00283-016-9673-y
0343-6993
1866-7414
wos:2017
WOS:000397962700005
Paycha, S (reprint author), Univ Potsdam, Golm, Germany., paycha@math.uni-potsdam.de
2022-06-21T08:22:50+00:00
sword
importub
filename=package.tar
34a3dc2b9807b6df5c7f4b241b26851a
Paycha, Sylvie
false
true
Sylvie Paycha
Mathematik
Institut für Mathematik
Referiert
Import
55331
2017
2017
eng
537
571
35
3
166
article
Duke Univ. Press
Durham
1
2016-11-09
2017-02-15
--
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.
Duke mathematical journal
10.1215/00127094-3715303
0012-7094
1547-7398
wos:2017
WOS:000394198200004
Guo, L (reprint author), Jiangxi Normal Univ, Dept Math, Nanchang, Jiangxi, Peoples R China.; Guo, L (reprint author), Rutgers State Univ, Dept Math & Comp Sci, Newark, NJ 08901 USA., liguo@rutgers.edu; paycha@math.uni-potsdam.de; zhangbin@scu.edu.cn
2022-06-24T13:15:02+00:00
sword
importub
filename=package.tar
9886193b5956a3b0f059d0000485446e
false
true
Li Guo
Sylvie Paycha
Bin Zhang
Mathematik
Institut für Mathematik
Referiert
Import
54085
2018
2018
eng
23
28
6
7
article
Springer
Cham
1
2018-09-27
2018-09-27
--
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.
Sustainable Futures for Higher Education : the Making of Knowledge Makers
10.1007/978-3-319-96035-7_2
978-3-319-96035-7
2364-6799
978-3-319-96034-0
wos:2018
WOS:000456958400004
Paycha, S (reprint author), Univ Potsdam, Potsdam, Germany., paycha@math.uni-potsdam.de
2022-02-28T09:16:39+00:00
sword
importub
filename=package.tar
a95f52859fd92ef03454a02a40a28488
false
true
Sylvie Paycha
Mathematik
Institut für Mathematik
Import
49226
2019
2019
eng
356
394
39
2
5
article
Springer
Cham
1
2019-06-05
2019-06-05
--
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.
European Journal of Mathematics
10.1007/s40879-018-0255-8
2199-675X
2199-6768
wos:2019
WOS:000466920800004
Paycha, S (reprint author), Univ Potsdam, Inst Math, D-14469 Potsdam, Germany.; Paycha, S (reprint author), Univ Clermont Auvergne, Lab Math Blaise Pascal, F-63001 Clermont Ferrand, France., clavier@math.uni-potsdam.de; liguo@rutgers.edu; paycha@math.uni-potsdam.de; zhangbin@scu.edu.cn
Natural Science Foundation of ChinaNational Natural Science Foundation of China [11521061, 11771190]; German Research Foundation (DFG)German Research Foundation (DFG) [PA 1686/6-1]
2021-02-03T09:01:10+00:00
sword
importub
filename=package.tar
c737e14149bcf6ca3b3dab09f066a2e6
Paycha, Sylvie
false
true
Pierre J. Clavier
Li Guo
Sylvie Paycha
Bin Zhang
eng
uncontrolled
Locality
eng
uncontrolled
Renormalisation
eng
uncontrolled
Algebraic Birkhoff factorisation
eng
uncontrolled
Partial algebra
eng
uncontrolled
Hopf algebra
eng
uncontrolled
Rota-Baxter algebra
eng
uncontrolled
Multivariate meromorphic functions
eng
uncontrolled
Lattice cones
Mathematik
Institut für Mathematik
Referiert
Import
Green Open-Access
44476
2020
eng
85
132
bookpart
European Mathematical Society Publishing House
Zürich
1
--
--
--
Renormalisation and locality
Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA)
Volume 2
branched zeta values
978-3-03719-205-4 print
978-3-03719-705-9 online
10.4171/205
Pierre J. Clavier
Li Guo
Sylvie Paycha
Bin Zhang
Mathematik
Institut für Mathematik
58767
2020
2020
eng
6185
6226
42
9
373
article
American Mathematical Society
Providence, RI
1
2020-07-08
2020-07-08
--
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.
Transactions of the American Mathematical Society
10.1090/tran/8067
0002-9947
1088-6850
outputup:dataSource:WoS:2020
WOS:000574902200005
Azzali, S (corresponding author), Univ Hamburg, Fachbereich Math Anal & Differentialgeometrie, Bundesstr 55, D-20146 Hamburg, Germany., sara.azzali@uni-hamburg.de; paycha@math.uni-potsdam.de
DFG German Research Foundation (DFG) European Commission
Azzali, Sara
2023-04-13T06:21:45+00:00
sword
importub
filename=package.tar
b77b70f911b10adb0f1ee486732afc0e
1474637-2
208386-3
false
true
Sara Azzali
Sylvie Paycha
Mathematik
Institut für Mathematik
Referiert
Import
61004
2020
2020
eng
19
2
61
article
American Institute of Physics
College Park, Md.
1
2020-02-03
2020-02-03
--
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.
Journal of mathematical physics
10.1063/1.5116381
0022-2488
1089-7658
outputup:dataSource:WoS:2020
022301
WOS:000518630800001
Guo, L (corresponding author), Rutgers State Univ, Dept Math & Comp Sci, Newark, NJ 07102 USA., clavier@math.uni-potsdam.de; liguo@rutgers.edu; <br /> paycha@math.uni-potsdam.de; zhangbin@scu.edu.cn
Natural Science Foundation of ChinaNational Natural Science Foundation; of China (NSFC) [11521061, 11771190, 11890663, 11821001]; German; Research Foundation (DFG)German Research Foundation (DFG) [FOR 2402]
Guo, Li
2023-10-05T08:41:03+00:00
sword
importub
filename=package.tar
8c8ceb529ff60616b45df3bf74577549
1472481-9
219135-0
2087956-8
false
true
Pierre Clavier
Li Guo
Sylvie Paycha
Bin Zhang
Mathematik
Institut für Mathematik
Referiert
Import
62393
2022
2022
eng
719
761
43
3
50
article
Springer
Singapore
1
2022-06-23
2022-06-23
--
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.
Vietnam journal of mathematics
10.1007/s10013-022-00570-7
2305-221X
2305-2228
outputup:dataSource:WoS:2022
WOS:000814978400001
Friz, PK (corresponding author), Tech Univ Berlin, Anton-Wilhelm-Amo-Straße 39, D-10117 Berlin, Germany.; Friz, PK (corresponding author), Weierstr Inst Angew Anal & Stochast, Anton-Wilhelm-Amo-Straße 39, D-10117 Berlin, Germany., bellinge@math.tu-berlin.de; friz@math.tu-berlin.de; <br /> paycha@math.uni-potsdam.de; preiss@math.uni-potsdam.de
DFG Research Unit [FOR2402]; European Research Council (ERC) under the; European Union [683164]
Friz, Peter K.
2024-01-31T13:55:16+00:00
sword
importub
filename=package.tar
9d434c383a6c7bb31e5d1123bfa5f93d
1481450-X
1319054-4
CC-BY - Namensnennung 4.0 International
Carlo Bellingeri
Peter Friz
Sylvie Paycha
Rosa Lili Dora Preiß
eng
uncontrolled
Signatures
eng
uncontrolled
Rough paths
eng
uncontrolled
Cartan's development
eng
uncontrolled
Renormalization
Mathematik
Institut für Mathematik
Referiert
Import
Hybrid Open-Access