Research Interests

My research interests are the frontier of Complex Analysis, Hypercomplex analysis, Operator Theory, and Functional Analysis. More precisely, I am interested in the following topics:

• Approximation theory:
  • Polynomial approximations in functions spaces (de Branges-Rovnyak spaces, Weighted Dirichlet spaces, Bergman spaces).
  • Approximations by kernels in reproducing kernel Hilbert spaces (RKHS).
  • Mathematics of machine learning.
• Geometric Function Theory:
  • Characterization of conformal welding.
  • Polynomial Lemniscates and conformal welding.
  • Holomorphic Dynamics.
• Hypercomplex Analysis:
  • Multicomplex analysis.
  • Dynamics in higher dimensions (Mandelbrot set, Julia sets).
  • Generation of 3D fractals.

I am currently:

• a member of the Laboratoire de mathématiques et physique fondamentales (Mathematics and fundamental physics research group).
• a research associate of the Laboratoire d'Intelligence Artificielle Appliquée (Applied artificial Intelligent research group).

Table of Contents

• Submitted articles
• Published articles
• Masters' Students
• Undergraduate Students
• Academic Documents
• Softwares

Submitted articles

• N. Doyon, P.-0. Parisé & W. Verrault. Counting Involutions on Multicomplex Spaces.
  Submitted in July 2025. [ArXiv]
• K. Lazebnik, P.-0. Parisé & M. Younsi. Rational Lemniscates and the Matching Problem.
  Submitted in June 2025. [PDF]
• Q. Charles & P.-O. Parisé. Classification of Principle 3D Slices of Filled-in Julia Sets in Multicomplex Spaces.
  Submitted in Mai 2025. [ArXiv]

Published articles (Total: 12)

• 2025:
  • J.-S. Dessureault, R. Lamontagne, P.-O. Parisé. The ethics of Creating Artificial Superintelligence: A Global Risk Perspective.
    AI and Ethics, published online. (2025) [ Open Access]
• 2024:
  • P.-O. Parisé & T. Ransford. On the Divergence of Taylor Series in de Branges-Rovnyak Spaces.
    PAMS, Series B, Vol. 11, pp.126-132. (2024) [Open Access]
  • P.-O. Parisé. Kernel-Summability Methods and the Silverman-Toeplitz Theorem.
    Recent progress in function theory and operator theory, AMS Contemporary Mathematics series, 799. (2024). [DOI, arXiv]
• 2022:
  • J. Mashreghi, P.-O. Parisé & T. Ransford. Power-series summability methods in de Branges-Rovnyak spaces.
    Integral Equations and Operator Thoery, 94 (20). (2022) [ DOI, arXiv]
  • P. Drouin & P.-O. Parisé. Un problème de rendez-vous.
    Bulletin de l'AMQ, Vol. 61, no. 4, 14p. (2022)
• 2021:
  • J. Mashreghi, P.-O. Parisé & T. Ransford. Failure of approximation of odd functions by odd polynomials.
    Constr Approx., Published online. (2021) [arXiv]
  • J. Mashreghi, P.-O. Parisé & T. Ransford. Cesàro summability of Taylor series in weighted Dirichlet spaces.
    Complex Analysis and Operator Theory, Vol. 15 (7). (2021) [DOI, arXiv]
• 2019:
  • G. Brouillette, P.-O. Parisé & D. Rochon. Tricomplex Distance Estimation for Filled-in Julia Sets and Multibrot Sets.
    IJBC, Vol. 29 (6), 15pp. (2019) [arXiv]
• 2017:
  • P.-O. Parisé. Dans l'imaginaire de Berhnard Riemann.
    Bulletin de l'AMQ, Vol. 57, no. 1, pp. 33-50. (2017)
  • P.-O. Parisé & T. Ransford & D. Rochon. Tricomplex Dynamical Systems Generated by Polynomial of Even Degree.
    CMSIM, Vol. 1, pp.38-49. (2017) [arXiv]
  • P.-O. Parisé & D. Rochon. Tricomplex Dynamical Systems Generated by Polynomials of Odd Degree.
    Fractals, Vol. 25 (3), 11 pp. (2017) [arXiv]
• 2015:
  • P.-O. Parisé & D.Rochon. A Study of Dynamics of the Tricomplex Polynomials ${\eta }^p+c$.
    Non. Lin. Dyn., Vol. 82 (1), pp.157-171 (2015) [arXiv]

Masters' Students

• 2025-: Sadio Cisse.
  Project title: Clustering and Deep Clustering: Applications in Artificial Intelligent.
  Financial Support: TBA.
  Co-supervisor: Nadia Ghazzali.

Undergraduate Students

• 2025: Stéphanie Couture.
  Project title: Gibbs phenomena and summability theory.
  Financial Support: NSERC Undergraduate Student Research Awards.
  Research report: In progress.
• 2024: Quentin Charles.
  Project title: Multicomplex Numbers and 3D slices of Filled-in Julia Sets.
  Financial Support: Undergradate Research Opportunities Program (UROP).
  Research report: submitted.
• 2022: Trey Summers (Co-supervision).
  Project title: Spectral Analysis of Tsunamis in Hawaii, Using Real Time Observations From the Pacific Islands Ocean Observing System.
  Financial Support: None, part of the student's mendatory project for graduating from the Oceanography program.
  Research report: Accepted to the journal Oceanography.
• 2021: Jérôme Côté.
  Project title: The Cesàro method applied to the Mandelbrot set.
  Financial Support: Institut des Sciences Mathématiques (ISM).
  Research report (In French): Jerome Cote research report.
• 2020: Philippe Drouin.
  Project Title: Rendezvous Problems on symmetric complete graphs.
  Financial Support: Institut des Sciences Mathématiques (ISM).
  Research report: Work done during this project was published in the Bulletin de l'AMQ (see the published articles section above).

Thesis

• PhD Thesis (in French): Sommabilité du développement de Taylor dans les espaces de Banach de fonctions holomorphes
• Pre-Doctoral exam research document (in French): Density of the disk algebra in de Branges-Rovnyak spaces.
• Master thesis (in French): Tricomplex Mandelbrot sets generalized to polynomials ${\eta }^p+c$.

Software

• Multibrot Voyager: github repository. You can also download the java application here.
• 3D Mandelbrot Voyager: github repository. You can download the java application here.
• Rational lemniscates: Link to the app.

Miscellaneous

• Involutions of Bicomplex Numbers: Preliminary work on involutions of Bicomplex numbers (2022).
• Fractal Achemy - Metatronbrot: Video realized with the software 3D Mandelbrot Voyager showing transitions between different 3D slices of the Tricomplex Mandelbrot set. Created by P.-O. Parisé & D. Rochon (2017). Uploaded on the YouTube platform and video has 112K views.