The evolution of systems, whether discrete or continuous, describes how the state of a system changes over time, often governed by equations like PDEs. In continuous systems, PDEs model and their discrete approximations model the evolution in both, time and space, with solutions depending on initial and boundary conditions.
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Simulation of a particle system influenced by a wave. Wave-type equations are understood using Fourier Integral Operators, the main field explored in my recent projects, along with harmonic analysis techniques, pseudo-differential operators and control theory in the analysis of PDE.
Oberwolfach Leibniz Project F2511 (2026) -
Harmonic analysis on compact homogeneous manifolds. Funded by MFO:
Mathematisches Forschungsinstitut Oberwolfach/
Oberwolfach Research Mathematics Institute
Germany>
Oberwolfach Leibniz Project F2511 (2026) -
Harmonic analysis on compact homogeneous manifolds. Funded by MFO:
Mathematisches Forschungsinstitut Oberwolfach/
Oberwolfach Research Mathematics Institute
Germany>
Project: Harmonic analysis on compact homogeneous manifolds (2026).
About the project: This project seeks to make a contribution in the field of Harmonic Analysis and its applications to Partial Differential Equations (PDEs). We want to investigate key unsolved issues appearing systematically in both pseudo-differential operators and in the approximation of these operators to parabolic partial differential equations. This project pursues two objectives: 1) to contribute with new theoretical results about pseudo-differential operators (PDOs) on compact Lie groups with applications to elliptic problems, and 2) to develop the application of the proposed approach to parabolic PDEs.
Keywords: Fourier Integral Operators, Microlocal Analysis, Control Theory, Harmonic Analysis, Partial Differential Equations.
Reference:
Cardona, D., Kumar, V., Ruzhansky, M. Pseudo-differential operators on homogeneous vector-bundles over compact homogeneous manifolds, submitted. arXiv:2403.08990.
Link
AIMS Research Project (2026):
Pseudo-differential operators and microlocal analysis. Funded by:
African Institute for Mathematical Sciences (AIMS South Africa).
South Africa>
Project: Pseudo-differential operators and microlocal analysis (2026). Funding: R 5000.
Role: Supervisor. Graduate Student: Mannondé Gbaguidi.
Analysis and PDE in Developing Countries (2023–Now) – ICMAM Latin America
Analysis and PDE in Developing Countries (2023–Now) – ICMAM Latin America
Project: Analysis and PDE in Developing Countries (2023–Now).
Institution: ICMAM Latin America (International Community of Mathematicians from Latin America).
Role: Principal Investigator.
Project Website: Website of the project .
Description: The project “Analysis and PDE in Developing Countries” of ICMAM Latin America aims to provide a broad perspective on mathematical research in Latin America and other developing regions. This project includes a series of conferences in Analysis and PDE, as well as a series of ICMAM volumes dedicated to showcasing the research carried out by junior and senior mathematicians in these areas.
FWO Postdoctoral Project 1204824N (2023-2026) - Fourier Integral Operators. Funded by the Belgian Research Foundation FWO
Belgium>
FWO Postdoctoral Project 1204824N (2023-2026) - Fourier Integral Operators. Funded by the Belgian Research Foundation FWO
Belgium>
Project: Fourier Integral Operators on Graded Lie groups (2023-2026)
Supported by: The Belgian Research Foundation - FWO.
Project Website: Visit the project page. Promotor: Prof. Dr. Michael Ruzhansky.
About the project: The goal of this project is to contribute to the scientific program carried out for more than forty years (dating back to the works of Folland and Stein) in charge of extending the techniques from the Euclidean harmonic analysis to the more general setting of nilpotent Lie groups. The contribution of this project will be the construction of the theory of Fourier Integral Operators (FIOs) on graded Lie groups. Developing a suitable theory of Fourier Integral Operators (FIOs) in this context will enable the solution of numerous unresolved problems in non-commutative harmonic analysis.
Project Scope: The scope of this project involves the collaboration between the Ghent Analysis and PDE Center with a team formed by researchers from several foreign universities, including UCLA, University of Chicago, Cornell University, University of Wisconsin Madison, Queen Mary University of London, and Imperial College London.
Keywords: Fourier Integral Operators, Microlocal Analysis, Control Theory, Harmonic Analysis, Partial Differential Equations.
Controllability of the fractional heat equation on compact manifolds (funded by the ERC Dycon Project # 694126)
Spain>
Controllability of the fractional heat equation on compact manifolds (funded by the ERC Dycon Project # 694126)
Spain>
Project: Controllability of the fractional heat equation on compact manifolds (April-July 2022)
Supported by: ERC (European Research Council) DyCon (Dynamic Control and Numerics of Partial Differential Equations) Project.
Website of DyCon: Visit the project page. Supervisor: Prof. Dr. Enrique Zuazua. Website of the project: Visit the project page.
About the project: Using the theory of pseudo-differential operators, in this project we have analised the null-controllability of the fractional heat equation associated to an elliptic positive pseudo-differential operator of positive order. We follow the methodology developed by Lebeau and Robbiano in the 90’s, using the technique of the spectral inequalities for wave packets of eigenfunctions. Then, using the pseudo-differential calculus on compact manifolds (without boundary), we derive the corresponding spectral inequality for any elliptic (pseudo-) differential operator of positive order. The obtained spectral inequality allows us to deduce the null-controllability of its corresponding heat equation.
Institutions: Deusto Institute of Science and Technology, Bilbao, Basque Country.
Keywords: Control Theory, Microlocal Analysis, Geometric Analysis.
Global Minds Project (GMF) 2026: Global Minds in PDE and Control Theory
Global Minds Project (GMF) 2026: Global Minds in PDE and Control Theory
Project Title: Global Minds in PDE and Control Theory
Collaboration: Belgium–Brazil.
Description: It is our pleasure to announce that the Ghent Analysis and PDE Center has been awarded one of the Global Minds Fund (GMF) International Grants for 2026, funded by the Research Department at Ghent University. This initiative aims to address global challenges through innovative research and dynamic collaboration. This marks our second successful Global Minds cooperation project. The project focuses on cooperation with leading Brazilian mathematicians, aiming at educational advancement in Brazil. It seeks to provide younger generations with a strong perspective in partial differential equations and control theory through collaboration with distinguished partners.
Partners from Ghent University:
Prof. Dr. Michael Ruzhansky (PI), Ghent University.
Dr. Duvan Cardona (Co-PI), Ghent University.
Dr. Marianna Chatzakou (Co-PI), Ghent University.
Global South partners:
Prof. Dr. Jaqueline Godoy Mesquita (Universidade Estadual de Campinas, Brazil).
Prof. Dr. Paulo L. Dattori da Silva (Universidade de São Paulo, Brazil).
Prof. Dr. Alexandre Kirilov (Universidade Federal do Paraná, Brazil).
Prof. Dr. Marcelo R. Ebert (Universidade de São Paulo, Brazil).
ICMAM Latin America (International Community of Mathematicians from Latin America).
Value of the grant: €10,000
Global Minds Project (GMF) 2025: Global Minds in Pseudo-Differential Analysis
Global Minds Project (GMF) 2025: Global Minds in Pseudo-Differential Analysis
Project Title: Global Minds in Pseudo-Differential Analysis
Collaboration: Belgium-Colombia.
Website of the project: Visit the project page.
Description: This transformative project focuses on enhancing educational opportunities in Colombia by partnering with prominent Colombian mathematicians. The aim of this project is to impart significant knowledge in the field of pseudo-differential operators to the younger generation. This initiative is part of the Ghent Analysis and PDE Center’s commitment to supporting mathematical research in developing countries, particularly in Latin America.
Partners from Ghent University:
Prof. Dr. Michael Ruzhansky (PI), Ghent University.
Dr. Duvan Cardona (Co-PI), Ghent University.
Dr. Marianna Chatzakou (Co-PI), Ghent University.
Global South partners:
Prof. Dr. Jairo Hernández (Universidad del Norte, Colombia).
Prof. Dr. Bienvenido Barraza (Universidad del Norte, Colombia).
Prof. Dr. Carolina Neira Jiménez (National University of Colombia).
Prof. Dr. Liliana Posada (Universidad del Valle, Colombia).
ICMAM Latin America (International Community of Mathematicians from Latin America).
Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups (funded by the Ghent Analysis and PDE Center)
Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups (funded by the Ghent Analysis and PDE Center)
Project: PhD Thesis. Main topic: Subelliptic pseudo-differential operators and Fourier integral operators on compact Lie groups (2019-2023)
Supported by the Ghent Analysis and PDE Center
Website of the center: Visit the GAPDE Center. Promotor: Prof. Dr. Michael Ruzhansky. Publications related to this project are available on the website: Publications (2019-2023).
About the project: This dissertation develops a pseudo-differential operator theory adapted to an arbitrary Hörmander sub-Laplacian on compact Lie groups, or, in other words, associated with an arbitrary sub-Riemannian structure over a compact Lie group. By following recent developments in the field, namely, the global quantisation theories of pseudo-differential operators, one in compact Lie groups (due to M. Ruzhansky and V. Turunen) and the other one on graded Lie groups (as developed by M. Ruzhansky and V. Fischer), this dissertation solves several open problems that arose in this setting in (1) harmonic analysis, (2) spectral theory, (3) control theory, and (4) PDE, considered as different but related areas of modern mathematics. The problems considered here include: (i) the Lp-theory of these pseudo-differential classes in the sub-Riemannian setting and in the setting of graded Lie groups, (ii) the spectral and qualitative properties of operators in ideals of compact operators, (iii) the extension of the theory of oscillating singular integrals due to Fefferman and Stein on compact and on graded Lie groups, (iv) an analysis of the dyadic maximal function on graded Lie groups and (v) an extensive number of applications to the well-posedness of diffusions models on compact Lie groups including applications of our pseudo-differential techniques to control theory.
Institutions: Ghent University, Belgium.
Keywords: Microlocal Analysis, Geometric Analysis.
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