"Our ambition is to demonstrate links between problems that are different in principle"

  • BCAM participates in the project “Spectral Theory and PDE: Real and Fourier Analysis” of Dr. Luca Fanelli and Dr. Renato Lucà, principal investigators


Dr. Luca Fanelli and Dr. Renato Lucà are the principal investigators of the project “Spectral Theory and PDE: Real and Fourier Analysis” (Ec & Spec) of the Knowledge Generation Projects 2021 of the State Research Agency.

The project “Spectral Theory and PDE: Real and Fourier Analysis” (Ec & Spec) focuses on problems in Quantum Mechanics (especially related to the Schrödinger and Dirac equations) and magnetohydrodynamics that can be studied using fine harmonic analysis techniques. More specifically, the main objectives are the investigation of the spectral properties of relativistic Hamiltonians, the stability of breathing, oscillatory evolving soliton-like waves, and on the other hand, the point properties of the solutions of the PDEs of quantum and fluid mechanics, such as their convergence to the initial data and the persistence of the relevant topological structures. “The main goal is to bring the most sophisticated and recent mathematical techniques to the service of applications in these areas,” explains Dr. Luca Fanelli.

“My more ambitious goal is to extend some important results on the point-like behavior of solutions of the Schrödinger equation to the environment of compact varieties (instead of the whole space). This requires an understanding of the phenomenon of resonance and interference, which provides fundamental insights into the nature of this important equation,’ says Dr. Renato Lucà.

For Fanelli, ‘the most interesting aspect of this project is undoubtedly the “transversality” of its objectives. The four main topics come from very different areas of applied mathematics, yet they share the basic technique, and our ambition is to demonstrate links between problems that are in principle different.

In the Ec & Spec project, they will research from a mathematical point of view the behavior of solutions that are fundamental to physicists, “we hope that our research will have an impact on our understanding of some important physical phenomena,” comments Lucà. “For example, we will study (mathematically) the reconnection (topology change) of magnetic fields associated with plasmas (MHD equation). Magnetic reconnection is a mechanism proposed by physicists to explain the phenomenon of solar flares, which is far from being fully understood. We believe that our research will provide more information in this regard.”

For Dr. Fanelli on the project, “the international component will allow us to introduce points of view that can bring a lot to the local scientific community. Our aim will then be to disseminate the results among scientists from various fields (in particular physicists, chemists, and geologists), to start collaborations at an interdisciplinary level”.

The news of being selected for the State Research Agency’s Knowledge Generation Projects 2021 is very important for the principal investigators. “It allows us to identify ourselves within the national scientific community, to start forming a “research group” and to give our contribution to research in applied mathematics, which has more and more impact on our lives. In addition to increasing the experience and maturity in the scientific supervision of young researchers,” they say.

As for the next challenges in the future, Dr. Fanelli and Dr. Lucà explain that there are two, “first, is to achieve all the objectives of this project. To lead a group of young and motivated researchers who, with the support of this grant, as well as the international visibility and support of BCAM, will facilitate this opportunity. The second is to discover connections so that the end date of the project represents a new beginning.”

About the principal investigators

Dr. Luca Fanelli is Ikerbasque Research Associate, UPV/EHU Linear and Non-Linear Waves.  He graduated in mathematics from the University of Bari in Italy in 2003. In 2008, he obtained his Ph.D. at the University of Rome La Sapienza. During his Ph.D., he was at the University of the Basque Country – UPV/EHU, where he started his scientific collaboration with Luis Vega. In 2009, he obtained the Spanish national Juan de la Cierva grant and a year later the Ramón y Cajal grant, both at the UPV/EHU. Since 2018 he has been a professor at the University of Rome La Sapienza, where he started in 2011 as a permanent researcher. His research interests are the dispersive PDEs of quantum mechanics, later opening up to Harmonic Analysis and more recently spectral theory. Throughout his career, he has supervised two Ph.D. theses, 18 Master’s theses, and 29 Bachelor’s theses.

Dr. Renato Lucà works in the mathematical field of harmonic analysis and its applications to partial differential equations. This involves, broadly speaking, the study of functions and solutions of equations by decomposing them as a sum of simple objects called harmonic waves. He received his Ph.D. from the University of Rome La Sapienza. He was a postdoctoral researcher at the Institute of Mathematical Sciences – ICMAT in Madrid and at the University of Basel in Switzerland. Since September 2019 he is Ikerbasque Research Fellow in the Linear and Non-Linear Waves research line at the Basque Center for Applied Mathematics – BCAM.