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.

When I arrived at UQTR, I was asked to get involved in some of the active research groups there. 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
• Accepted articles
• Undergraduates' projects supervision
• Academic Documents
• Softwares

Submitted articles

• N. Doyon, P.-0. Parisé & William Verrault. Counting Involutions on Multicomplex Spaces.
  Submitted on March 2024. [preprint]

Accepted articles

• Anna Laure Gagné-Landmann, Polina Khapikova, Yujie Wang, A. Anthony Bloom, John R. Worden, Valeria Barra, Renato Kerches Braghiere, Christian Frankenberg, Pierre Gentine, Tapio Schneider, Katherine Deck, Pierre-Olivier Parisé. An improved plant hydraulics model for the land component of the CliMA Earth System Model.
  Accepted in the Proceedings of AGU Fall Meeting, July 2022.

Published articles (Total: 11)

• Spaces of Holomorphic Functions and Summability Theory:
• 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]
• 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]
• 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]
• Hypercomplex Structures:
• 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]
• 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]
• 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]
• Game Theory:
• P. Drouin & P.-O. Parisé. Un problème de rendez-vous.
  Bulletin de l'AMQ, Vol. 61, no. 4, 14p. (2022)
• Miscellaneous:
• P.-O. Parisé. Dans l'imaginaire de Berhnard Riemann.
  Bulletin de l'AMQ, Vol. 57, no. 1, pp. 33-50. (2017)

Undergraduates' Projects Supervision

• 2024: Quentin Charles.
  Project title: Multicomplex Numbers and 3D slices of Filled-in Julia Sets.
  Financial Support: Undergradate Research Opportunities Program (UROP).
  Research report: In progress.
• 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: Submitted 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.

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.