Pierre-Louis Giscard

Maître de Conférences (Lecturer), Algebraic-Combinatorics
Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA)
Université Littoral Côte d'Opale, Calais

 Tel: +33 3 21 46 55 84
Email: giscard@univ-littoral.fr

Research interests

 My research is centered on the combinatorial and algebraic structures associated with walks (a.k.a. paths) on graphs and their applications throughout mathematics and physics. My research in algebra focuses on structures associated with graph-walks and their use, notably for solving non-autonomous differential systems in physics. On the combinatorics front, my long term goal is to approach the long standing open problem of enumerating self-avoiding polygons on infinite regular lattices via a completely algebraic route, by extending techniques from number theory to walks. I am always keen on discussing research and participating in research collaborations on a wide variety of subjects, from quantum mechanics to applied network analysis.

I am the coordinator of the ANR (French National Research Agency) JCJC research project ALCOHOL (ALgebraic COmbinatorics of Hikes On Lattices), 2019-2024
I am one of the coordinators of the ANR research project MAGICA (MAGnetic resonance techniques and Innovative Combinatorial Algebra), 2021-2024
I am one of the participants of MathAmSud project ALGonCOMB (Algebraic structures supported on families of combinatorial objects), 2022-2025

Keywords : posets, Cartier-Foata monoids, self-avoiding walks, self-avoiding polygons, Hopf algebras
ORCID : 0000-0003-3025-8750
ResearcherID : HPH-1942-2023
ResearchGate : here
Google scholar : here
Academia : here
Team Current research team and open positions:
Manon Ryckebusch, PhD student.
Omid Faizy Namarvar, postdoctoral researcher, see project description.
Yohan Hosten, postdoctoral researcher, see project description.

Past members:
Théo Karaboghossian, postdoctoral researcher 01/2021-01/2022

Publications

Publications in journals
All are available on the arXiv in open access

  • Algebraic Walk Theory
    P.-L. Giscard
    Report, Habilitation à Diriger des Recherches
    Open access link (2023)

  • A co-preLie structure from chronological loop erasure in graph walks
    L. Foissy, P.-L. Giscard, C. Mammez
    Preprint
    Open access link (2023)

  • Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory
    P.-L. Giscard, M. Foroozandeh
    Computer Physics Communications, 2022, p. 108561, doi: 10.1016/j.cpc.2022.108561
    Journal link, open access link (2022)

  • Realizable cycle structures in digraphs
    J. Fromentin, P.-L. Giscard, T. Karaboghossian
    European Journal of Combinatorics, Vol. 113, 103748 (2023), doi: 10.1016/j.ejc.2023.103748. Journal link, open access link (2023)

  • Computations of general Heun functions from their integral series representations
    T. Birkandan, P.-L. Giscard, A. Tamar
    2021 Days on Diffraction (DD), 2021, pp. 12-18, doi: 10.1109/DD52349.2021.9598600.
    Journal link, open access link (2021)

  • Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
    P.-L. Giscard, A. Tamar
    Editor's pick at Journal of Mathematical Physics, Vol.63, Issue 6: 063501 (2022).
    Journal link, open access link.

  • Asymptotic counts for the walk multiples of self-avoiding polygons on lattices using sieves
    P.-L. Giscard
    Discrete Mathematics 344(4): 112305, doi 10.1016/j.disc.2021.112305, (2021).
    Journal link, open access link.

  • Tridiagonalization of systems of coupled linear differential equations with variable coefficients by a Lanczos-like method
    P.-L. Giscard, S. Pozza
    Linear Algebra and Its Application 624: 153-173, (2021)
    Journal link, open access link.

  • Lanczos-like algorithm for the time-ordered exponential: the *-inverse problem
    P.-L. Giscard, S. Pozza
    Applied Mathematics, 1-21, doi 10.21136/AM.2020.0342-19, (2020).
    Journal link, open access link.

  • On the solutions of linear Volterra equations of the second kind with sum kernels
    P.-L. Giscard
    Journal of Integral Equations and Applications, 32(4): 429-455 (2020).
    Journal link, open access link.

  • A Lanczos-like method for non-autonomous linear ordinary differential equations
    P.-L. Giscard, S. Pozza
    Bollettino dell'Unione Matematica Italiana, 2198-2759 (2022).
    Journal link, open access link (2022)

  • Dynamics of quantum systems driven by time-varying Hamiltonians: Solution for the Bloch-Siegert Hamiltonian and applications to NMR
    P.-L. Giscard, C. Bonhomme
    Physical Review Research 2, 023081 (2020)
    Journal link, open access link

  • Computing Optimal Assignments in Linear Time for Approximate Graph Matching
    N. M. Kriege, P.-L. Giscard, F. Bause, R. C. Wilson
    2019 IEEE International Conference on Data Mining (ICDM), Beijing, China, 2019, pp. 349-358, (2019).
    Journal link, open access link.

  • A general purpose algorithm for counting simple cycles and simple paths of any length
    P.-L. Giscard, N. M. Kriege, R. C. Wilson
    Algorithmica 81 (7), 2716-2737 (2019)
    Journal link , open access link.

  • Enumerating simple paths from connected induced subgraphs
    P.-L. Giscard, P. Rochet
    Graphs and Combinatorics 34(6): 1197-1202, (2018).
    Journal link, open access link.

  • A Centrality Measure for Cycles and Subgraphs II
    P.-L. Giscard, R. C. Wilson
    Applied Network Science 3(1): 9, (2018).
    Journal link, open access link.

  • The All-Paths and Cycles Graph Kernel
    P.-L. Giscard, R. C. Wilson
    Open access link

  • Cycle-Centrality in Economic and Biological Networks
    P.-L. Giscard, R. C. Wilson
    Complex Networks & Their Applications VI (Cherifi, C., Cherifi, H., Karsai, M., Musolesi, M., eds.), Springer International Publishing, 14-28, (2018).
    Journal link, open access link.

  • A Hopf algebra for counting cycles
    P.-L. Giscard, P. Rochet, R. C. Wilson
    Discrete Mathematics, volume 341:1439-1448 (2018).
    Journal link, open access link.

  • Evaluating balance on social networks from their simple cycles
    P.-L. Giscard, P. Rochet, R. C. Wilson
    Journal of Complex Networks, volume 5: 750-775 (2017).
    Journal link, open access link.

  • On Valid Optimal Assignment Kernels and Applications to Graph Classification
    N. M. Kriege, P.-L. Giscard, R. C. Wilson
    Proceedings of the 30th International Conference on Neural Information Processing Systems, 1623-1631, (2016).
    Journal link, open access link.

  • Algebraic combinatorics on trace monoids: extending number theory to walks on graphs
    P.-L. Giscard, P. Rochet
    SIAM Journal on Discrete Mathematics, volume 31: 1428-1453, (2017).
    Journal link, open access link.

  • Exact Inference on Gaussian Graphical Models of Arbitrary Topology using Path-Sums
    P.-L. Giscard, Z. Choo, S. J. Thwaite, D. Jaksch
    Journal of Machine Learning Research, volume 17: 1-19 (2016).
    Journal link, open access link.

  • An Exact Formulation of the Time-Ordered Exponential using Path-Sums
    P.-L. Giscard, K. Lui, S. J. Thwaite, and D. Jaksch
    Journal of Mathematical Physics, 56(5):053503 (2015).
    Journal link, open access link.

  • A Graph Theoretic Approach to Matrix Functions and Quantum Dynamics
    P.-L. Giscard
    PhD Thesis, Keble College, University of Oxford, link (2014)
    Mathematical, Physical and Life Sciences division, University of Oxford Thesis Commendation, Thesis Prize

  • Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums
    P.-L. Giscard, S. J. Thwaite, D. Jaksch
    SIAM Journal on Matrix Analysis and Applications, 34(2):445-469 (2013).
    Journal link, open access link.

  • Walk-Sums, Continued Fractions and Unique Factorisation on Digraphs
    P.-L. Giscard, S. J. Thwaite, D. Jaksch
    Open access link (2012)

  • Optomechanical control of atoms and molecules
    M. Bhattacharya, S. Singh, P.-L. Giscard, P. Meystre
    Laser Physics, 20(1):57-67, (2010)
    Journal link.

  • Entangling the ro-vibrational modes of a macroscopic mirror using radiation pressure
    M. Bhattacharya, P.-L. Giscard, P. Meystre
    Phyical Review A, 77:030303, (2008)
    Journal link, open access link.

  • Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror
    M. Bhattacharya, P.-L. Giscard, P. Meystre
    Physical Review A, 77:013827 (2008).
    Journal link, open access link.

  • Long distance beam propagation in colloidal suspensions: comparison between theory and experiment
    E. M. Wright, W. M. Lee, P.-L. Giscard, K. Dholakia
    Proceedings of SPIE - The International Society for Optical Engineering Volume 7038, 70380P (2008).
    Journal link.

    Unpublished papers, mostly produced during my MSc:
  • An Explicit Bound for Dynamical Localisation in an Interacting Many-Body System, link
  • The walk-sum method for simulating quantum many-body systems, link
  • Tunable Supersolids of Rydberg Excitations Described by Quantum Evolutions on Graphs, link
  • Quantum Mechanical Limits to Inertial Mass Sensing by Nanomechanical Systems, link
  • Operators for the Aharonov-Anandan and Samuel-Bhandari Phases, link
  • Considerations about the Aharonov-Anandan Phase for Time Independent Hamiltonians, link
  • Optical squeezing of a mechanical oscillator by dispersive interaction, link

Abstracts published in peer-reviewed conferences:

  • Ultra-fast convergent analytical solutions of the Bloch equations: the Path-Sum approach
    C. Bonhomme, P.-L. Giscard
    62nd Experimental Nuclear Magnetic Resonance Conference (2021).

  • Undamped and Damped Bloch Equations: Exact Solutions by Path-Sum
    C. Bonhomme, P.-L. Giscard
    61st Experimental Nuclear Magnetic Resonance Conference, Baltimore, Maryland, (2020).

  • Theory of Solid state NMR: from Dyson, Magnus, Feynman to Path-Sum
    C. Bonhomme, P.-L. Giscard
    Alpine Conference on Magnetic Resonance in Solids (2019).

  • A New Approach of Ordered Exponential in NMR: the Path-Sum
    C. Bonhomme, P.-L. Giscard
    60th Experimental Nuclear Magnetic Resonance Conference, (2019).


Online articles for math popularization:
  • Que sait-on compter sur un graphe ? Partie 3: Matrices et chemins
    P.-L. Giscard, Image des Maths, link, December 2020

  • Que sait-on compter sur un graphe ? Partie 2: Motifs et polygones
    P.-L. Giscard, Image des Maths, link, October 2020

  • Que sait-on compter sur un graphe ? Partie 1: Des graphes du monde qui nous entoure
    P.-L. Giscard, Image des Maths, link, July 2020
Raw data & Algorithms Raw molecular data pertaining to the cationic tin oxo-cluster used in "General solutions for quantum dynamical systems driven by time-varying Hamiltonians: applications to NMR" arXiv 1905.04024, provided by Dr. F. Ribot, is available here
Molecular dynamics movies from the same work: Movie 1, Movie 2, Movie 3, Movie 4, Movie 5, Movie 6 and the colour bar

Mathematica notebook for the Bloch-Siegert Hamiltonian: here

Heun functions:
Pseudocode for the numerical evaluation of General Heun Functions: here
Python codes for the computation of general Heun functions: GitHub

Counting simple cycles and simple paths on graphs
MATLAB versions:
For counting simple cycles of any length on any (weighted directed) graph: CycleCount
For counting simple paths of any length on any (weighted directed) graph: PathCount
For counting both simple cycles and simple paths at the same time: CyclePathCount
PYTHON versions: here

Machine learning on structured datasets
All-cycles and paths kernel for automatic graph classification tasks: here
Weisfeiler-Lehman optimal assignment on graphs (MATLAB): here
Weisfeiler-Lehman optimal assignment on graphs (JAVA): here

Loop-centrality on large networks
Algorithms determining the asymptotic fraction of network flows passing through selected loops (MATLAB): here and there.

Seminars

 Future presentation(s) and visit(s):

  • Seminar on "Time-dependent Hamiltonians" at the workshop "Controlling Time-Dependent Aspects of Quantum Systems: from Theory to Application", Sorbonne University, Paris, 21st March 2024.
  • Invited speaker at "70 years of Magnus series" Conference, July 1-5, 2024, Univ. Jaume I, Castellón, Spain.


    Past presentations:

    • Defense, Habilitation à Diriger des Recherches, Algebraic Walk Theory, slides , ULCO, Calais, 15/12/2023.
    • Seminar Dénombrement des polygones auto-évitants: la route déterministe, CALIN, LIPN, Université Paris-Nord, Villetaneuse, 17/10/2023.
    • Prague Workshop on Numerical Mathematics, invited presentation and research stay, link, July 2023
    • Seminars by M. Ryckebusch on a joint work, Between graph theory and differential equations: the *-product, Amiens (March 2023) and Calais (May 2023), presentation and slides.
    • Seminar by T. Karaboghossian on a joint work, Why walks lead us astray in the study of graphs, Calais, 16th December 2021, slides.
    • Invited presentation, A Lanczos-like algorithm for time-ordered exponentials, 8th European Congress of Mathematics, Portorož, Slovenia, 21st June 2021, slides.
    • T. Birkandan, P.-L. Giscard, A. Tamar, A Python implementation of the general Heun function using elementary integral series, Contributed presentation Days on Diffraction 2021, May 31 – June 4, 2021 St. Petersburg, slides and recorded presentation, starting from 39:30.
    • Alcohol Workshop on walk combinatorics (WACA days), 25-28 May 2021
    • Contributed presentation by C. Bonhomme on joint research, Hydroxyapatite revisited by DNP and spin diffusion: Natural abundance 43Ca DNP MAS and ultra-high magnetic field (35T) 43Ca experiments, modeling and new mathematical treatment of DNP process, American Chemical Society Spring 2021 meeting, April 5-16 2021, slides
    • Contributed presentation by C. Bonhomme on joint research, Ultra-fast convergent analytical solutions of the Bloch equations: the Path-Sum approach, Experimental Nuclear Magnetic Resonance Conference ENC 2021, March 29-31 2021, slides
    • Invited seminar, Dénombrement des polygones auto-évitants: la route déterministe, Institut de Recherche Mathématique Avancée, Strasbourg, France, 18th February 2021, slides
    • Invited seminar, Unexpected applications of walk combinatorics, Laboratoire Paul Painlevé, Université de Lille, 31st January 2020, slides
    • Official presentation of the ANR ALCOHOL research project, Agence Nationale de la Recherche, Paris, 13th November 2019, slides
    • Invited Talk, Combinatoire des chemins. De la théorie des nombres à la mécanique quantique, Fédération de Recherche Mathématique du Nord Pas de Calais, 15th October 2019, slides
    • Short Talk by C. Bonhomme on a joint work, Theory of Solid State NMR: from Dyson, Magnus, Feynman to Path-Sum, Alpine Conference on Magnetic Resonance in Solids, 15-19th September 2019, slides
    • Contributed talk by Stefano Pozza (Charles University) on a joint work, Lanczos-like method for the time-ordered exponential, International Conference on Matrix Analysis and its Applications (MATTRIAD 2019), 9th September 2019, slides
    • Cambridge visit and meeting with Prof. Bonhomme, Prof. Pickard and Prof. Haydock; 15-17th July 2019
    • Stefano Pozza (Charles University), ULCO matrix analysis seminar on a joint work, Lanczos-like method for the time-ordered exponential, 25th June 2019, slides
    • Invited talk, Walk combinatorics: from number theory to quantum mechanics, ALEA Young 2019 workshop, 19-24 May 2019 Caen, slides
    • Contributed talk by C. Bonhomme on a joint work, A New Approach of Time-Ordered Exponential in NMR: the Path-Sum, 60th Experimental Nuclear Magnetic Resonance Conference, April 7-12, 2019, Pacific Grove, California, slides
    • Outreach, short presentation to high-schoolers Walking on graphs, 13 February 2019
    • Short scientific presentation Combinatoire algébrique des chemins, HCERES visit, 11 February 2019, slides
    • Invited seminar and short research stay, Networks analysis from number theoretic sieves : biological applications University of Strathclyde, 5-9 February 2019, slides
    • Seminar, Counting graph walks with sieves joint seminar University of Kent - Université du Littoral Côte d'Opale, 30 January 2019, slides
    • Seminars, Cribles et polygones auto-évitants I & II, Mathematics Department, Université du Littoral Côte d'Opale, 8 and 29 November 2018
    • Seminar, Une série pour la constante connective du réseau carré, Laboratoire d'informatique de l'École polytechnique (LIX), 21 November 2018
    • Contributed talk, Un crible non-commutatif en biologie, rencontres ALEA, CIRM Marseilles, March 2018, slides
    • Invited seminar, Théorie des chemins et informatique : interactions, Laboratoire d'Informatique Fondamentale, Université Aix-Marseilles, March 2018
    • Invited seminar, Des nouvelles de l'exponentielle ordonnée, Institut Mathématique de Bourgogne, Université de Bourgogne, March 2018, slides
    • Seminar, La théorie des chemins: une extension de la théorie de nombres, laboratoire Joseph Liouville, Université Littoral Côte d'Opale, January 2018
    • Extended visit to the University of Talca, Chile, group of Prof. María Ronco, November 2017.
    • Seminar, Introduction aux cribles : théorie des nombres & polygones auto- évitants, Laboratoire Bordelais de recherche en informatique (LaBRI), October 2017, slides
    • Contributed Talk, Loop Centrality in Complex Economic and Biological Networks, Sixth International Conference on Complex Networks and their Applications, Lyon, November/December 2017
    • Invited seminar, La théorie des chemins. Une extension de la théorie des nombres et ses applications, Team SPOC, Laboratoire IMB (Univ. Bourgogne, Dijon, France), 27th September 2017, slides
    • Seminar, The theory of walks. An overview, Max Planck Institute for Informatics, Saarbrucken, July 2017, slides
    • Short presentation, Loop centrality in complex networks, CoSyDy meeting, Leeds UK, 25th July 2017
    • Short presentation, Extending number theory to walks on graphs, Journees ALEA, Centre International de Rencontres Mathématiques (CIRM), Marseilles, France, 20-24 March 2017, slides
    • Invited seminar, Cycle Counting: Theory and Applications, Stochastic Processes Group, University College London, United Kingdom, 8th March 2017slides
    • International Summer School on Complex Networks, Bertinoro, Italy, 11-15th July 2016
    • Invited seminar, Network Cluster Lecture, Extending number theory to walks on graphs: why? how? and so what?, Keble College Advanced Studies Centre, Oxford, UK, 25th April 2016, slides
    • Invited seminar given by P. Rochet, Walks are just connected hikes, University of Antofagasta, Chile, 26th November 2015, slides
    • Seminar, Of walks and graphs: an introduction to walk theory, University of York, UK, 15th October 2015, slides
    • Invited seminars, Walk Theory I & II , University of Geneva, Switzerland, September 2014, slides: talk 1, talk 2
    • Invited seminar, Walk theory, with applications, Said Business School, Oxford, UK, 14th August 2014, slides
    • Network journal club seminar, Of Walks and Graphs, An Introduction to Walk Theory, Department of Mathematics, Oxford, UK, 16th July 2014, slides
    • Atomic and Laser Physics Seminar, Walk Theory, Oxford, UK, 9th June 2014, slides
    • Poster presented by S. Thwaite, Evaluating Matrix Functions by Resummations on Graphs: the method of walk-sums", Computational methods for quantum materials, Sherbrooke, Canada, June 2014, poster
    • Poster presentation, The method generating theorem, Workshop on Advances in Matrix Functions and Matrix Equations, University of Manchester, UK, March 2013, e-poster
    • Invited seminar, Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums, Numerical Analysis Group, Department of Mathematics, Oxford, UK, November 2012, slides
    • Invited seminar, Matrix functions as continued fractions, Numerical Linear Algebra Group, Department of Mathematics, University of Manchester, UK, October 2012, slides
    • Poster, The walk-sum and path-sum methods for simulating quantum many-body systems, International congress on mathematical-physics, Aalborg, Denmark, August 2012, e-poster
    • Contributed presentation, Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums, SIAM Conference on applied linear algebra, Valencia, Spain, June 2012, slides
    • Poster, Walk-sum, a new approach to quantum evolutions, Windsor workshop on dynamics and simulation of ultra-cold matter, Windsor, UK, August 2011
    • Atomic and Laser Physics Seminar, The Theory of Operator-Lattices A New Analytical Tool for Quantum Many-Body Physics, Oxford, UK, September 2010, slides
    • Poster, An Analytical Tool for Quantum Many-Body Physics, Les Houches physics summer school, Les Houches, France, July 2010
    • Presentation by E. Wright, Long distance beam propagation in colloidal suspensions: comparison between theory and experiment, SPIE Optics and Photonics, San Diego, USA, August 2008
    • Poster, Entanglement of a Laguerre-Gaussian cavity mode with a rotating mirror, DAMOP, Pennsylvania State College, USA, May 2008

    • Teaching / Enseignement

    		 A l'ULCO:
  • Cours de Maths 1, L1 MSPI, "Fonctions réelles d'une variable réelle"
  • Cours de Maths 2, L1 MSPI, "Théorie des ensembles"
  • Cours de Maths 3, L1 MSPI, "Nombres complexes et géométrie"
  • Cours de Maths 5, L1 MSPI, "Algèbre linéaire"
  • Cours de soutien en mathématiques pour les étudiants en difficulté
  • Organisation des tutorats d'accueil pour les primo-entrants en L1 SV et L1 MSPI
  • Cours de Mathématiques pour la Physique, L2 Physique-Chimie
  • Préparation aux écrits d'agrégation en mathématiques
  • Cours spécialisé M2 recherche, "Fonctions de matrices et graphes"
  • Cours d'anglais scientifique M1 recherche
  • Cours d'anglais scientifique M2 recherche

    Previously:

  • I was a supervisor for experimental optics (telemetry and infrared), Oxford (2010-2015)
  • I taught quantum mechanics and mathematical-physics to both mathematics and physics students in Keble and Somerville colleges, Oxford (2010-2015)
  • I was an interviewer at the Oxford interviews for applicants to the Mathematics course at Keble college (2014)
  • I taught Numerical Analysis at the University of York (2016-2017)

    • Short bio

    		I completed two MSc degrees from the University of Arizona and Institut d'Optique Graduate School in 2009 (Diplôme d'Ingénieur). That same year I was awarded a Scatcherd European scholarship to fund my DPhil in mathematical-physics from Keble College, Oxford, which I completed in 2014 under the direction of Prof. Dieter Jaksch. I received a University of Oxford Thesis Prize for my thesis "A graph theoretic approach to matrix functions and quantum dynamics".

    Following my DPhil, I held a postdoc position at Oxford funded by the successful EPSRC Grant EP/K038311/1 obtained together with Prof. Jaksch, Dr. Stephen Clark and Dr. Sarah Al-Assam. In 2015, I was awarded a three years Research Fellowship by the Royal Commission for the Exhibition of 1851 and chose to undertake it in the group of Prof. Richard Wilson, at the University of York.

    I am a lecturer (Maître de Conférences) of algebraic combinatorics at the Université du Littoral Côte d'Opale, since September 2018.
    My full CV is available here.