Exploring New Frontiers: Chronicle of the School on Homogenization and Fractional Calculus

On 24-25 March 2025, the Centre de Recerca Matemàtica (CRM) hosted the ‘School on New Frontiers in Homogenization and Fractional Calculus’ to celebrate the 50th anniversary of Γ-convergence. This mathematical technique was introduced by Ennio De Giorgi and Tullio Franzoni in 1975.

The event brought together leading experts in the fields of homogenization and fractional calculus to discuss the latest developments and applications of these mathematical techniques. Topics included phase transitions and asymptotic analysis of partial differential equations, as well as nonlocal problems, which are becoming increasingly relevant in materials science.

It also featured a series of lectures and discussions that highlighted the current state of research and provided valuable insights into the future direction of these fascinating areas of mathematics.

Alberto Maione during his presentation to the course.

In particular, the speakers were:

Andrea Braides, Full Professor at the University of Rome “Tor Vergata”. His research focuses on the calculus of variations, Γ-convergence, and homogenization. During the event, Braides gave a detailed overview about recent results on singular perturbations with fractional and higher-order seminorms, in the framework of phase transitions and of free-discontinuity problems.

Maicol Caponi, fixed-term Researcher at the University of L’Aquila. His research focuses on the calculus of variations, partial differential equations, and fracture dynamics, with applications in material science. In his talk, he spoke about recent results with collaborators, where the authors derived a strain-gradient theory for plasticity as the Γ-limit of discrete dislocation fractional energies, without the introduction of a core-radius.

Alessandro Carbotti, a Postdoc at the University of Salento, whose research focuses on fractional analysis, sub-Riemannian geometry and partial differential equations. Carbotti presented a new approach to the H-convergence problem involving nonlocal linear operators in fractional divergence form. If symmetry is also assumed, he showed that the H-compactness of the operators is equivalent to the Γ-convergence of the associated energies.

Patrick Dondl, Full Professor at the University of Freiburg (Germany), whose research focuses on numerical methods for partial differential equations and applications in mechanics. In his talk, he presented a numerical method for approximating the fractional Laplacian, approximating the operator using a basis formed by dilated and shifted tensor products of sinc functions.

Alex Ferrer, Associate Researcher at CIMNE and “Serra-Hunter” Associate Professor at Universitat Politècnica de Catalunya. His research focuses on structural optimization and additive manufacturing. In his talk, he presented the dehomogenization method for stress minimization in lattice structures, highlighting its efficiency compared to multi-scale topology optimization. Ferrer explained the use of homogenization to optimize macroscopic structures and addressed singularities in the orientation field, validating the methodology with numerical examples.

Carolin Kreisbeck during her talk.

Carolin Kreisbeck, Full Professor at the Catholic University of Eichstätt-Ingolstadt, whose research focuses on the calculus of variations and nonlinear partial differential equations, with applications in material science. Kreisbeck explored variational problems with integral functionals involving nonlocal gradients, focusing on Sobolev spaces, trace operators, and Poincaré inequalities. She established existence of minimizers for functionals with quasiconvex or polyconvex Lagrangians, depending on heterogeneous nonlocal gradients.

Fabio Paronetto, Associate Professor at the University of Padua. His research focuses on the asymptotic behavior of solutions of elliptic and parabolic partial differential equations. He talked about the existence and uniqueness of solutions of forward-backward parabolic equations, extending classical results and providing a G-compactness result for sequences of such operators.

Xavier Ros-Oton, ICREA Research Professor and Full Professor at the University of Barcelona, is known for the study of regularity theory for partial differential equations, integro-differential operators, and free boundary problems. Ros-Oton presented recent results on the regularity of solutions of the Boltzmann equation, a fundamental equation in statistical mechanics, which models the evolution of gas. He extended previous results by Imbert and Silvestre, demonstrating that if certain observables, such as mass and pressure, remain bounded, then the solutions are smooth.

Simone Verzellesi, a Postdoc at the University of Padua, whose research focuses on the calculus of variations and sub-Riemannian geometry. In his talk, he presented recent results for sequences of anisotropic integral functionals depending on Lipschitz continuous vector fields. He discussed about the properties of the anisotropic functional setting, and showed integral representation results for these functionals, as well as Γ-compactness results for sequences of anisotropic integral functionals.

Xavier Ros-Oton during his talk.

Researchers from the Research in Pairs programme solving problems together.

Overall, the School on New Frontiers in Homogenization and Fractional Calculus was a success, fostering collaboration and innovation to advance the understanding of these mathematical techniques.