Three ICM speakers headline the first CRM Faculty Colloquium

As highlighted by the CRM Director during the welcome remarks, the new colloquium aims to foster synergies between research groups, showcase recent advances, and strengthen internal scientific cohesion. In this context, the first talk was delivered by Xavier Tolsa (ICREA – UAB – CRM), titled Quantitative rectifiability, singular integrals, and harmonic measure. Tolsa, who will be a plenary speaker at this year’s ICM, introduced the foundations of quantitative rectifiability, a research area that links the fine geometry of sets with tools from harmonic analysis, such as square functions and singular integrals. Throughout his lecture, he revisited classical results while highlighting recent developments that have resolved deep problems related to harmonic measure, including the one-phase and two-phase cases, as well as new findings about its dimension. His talk allowed the audience to explore a highly active research field with connections ranging from analysis to potential theory and partial differential equations.

The second lecture, Serendipity strikes again, was delivered by Joaquim Ortega-Cerdà (UB – CRM), who has been invited to the Analysis section of the ICM. Ortega-Cerdà guided the audience through a fascinating journey connecting a classical problem from the Bell Labs in the 1960s with a recent result in Fourier analysis. The “fortunate accident” that enabled Landau, Slepian, and Pollack to advance signal theory reappears in a different context: the characterization of an extremal function that maximizes its value at 0 among bandlimited functions with norm equal to 1. In a joint work with Bondarenko, Radchenko, and Seip, the team achieves a complete identification of this function, after decades in which only numerical approximations were available. The key once again lies in an unexpected commutation of operators, an episode of mathematical serendipity that gives meaning to the title of the talk and captivated the audience.

The session concluded with a lecture by Xavier Cabré (ICREA – UPC – CRM), invited to the Partial Differential Equations section of the ICM, titled Elliptic PDEs: recent extensions on Hilbert’s 19th problem. Cabré offered a historical and modern overview beginning with Hilbert’s 1900 question on the regularity of minimizers of variational integrals, followed by the foundational contributions of De Giorgi and Nash in the 1950s, and leading to Simons’ work on minimal surfaces in 1968. From there, Cabré focused on recent progress on the regularity of stable solutions, a much broader class than absolute minimizers. He presented results obtained in the past decade, including joint work with Figalli, Ros-Oton, and Serra, as well as contributions from other researchers on stable minimal surfaces. He also highlighted several important open problems, such as regularity in and the fractional case, which remain major challenges in the field.

With this inaugural edition, the CRM Faculty Colloquium establishes itself as a valuable opportunity to share high-level research, strengthen connections, and showcase the work of the mathematical community linked to the Centre. The quality of the talks and the discussions that followed bode well for the future of this initiative, which will continue to gather researchers every quarter.