8th BCAM–EHU Summer School on Harmonic Analysis and PDEs: Energy Minimisation in Analysis and Discrete Geometry Takes Place in Bilbao

• The summer school took place at the Basque Centre for Applied Mathematics (BCAM) and the University of the Basque Country (UPV/EHU) in Bilbao from 29 June to 3 July, bringing together graduate students and early-career researchers with three internationally recognised specialists in harmonic analysis and discrete geometry.

The Basque Centre for Applied Mathematics – BCAM and the University of the Basque Country – UPV/EHU organised the eighth edition of the BCAM–UPV/EHU summer school on Harmonic Analysis and PDEs, known within the research community as HAPDE. This year's theme was "Energy Minimisation in Analysis and Discrete Geometry". 

Over five days, three intensive mini-courses unpacked different mathematical perspectives on this landscape, each revealing unexpected connections to harmonic analysis, Fourier methods and discrete geometry.

Dmitriy Bilyk of the University of Minnesota opened the school with a course on energy minimisation on the sphere, discrepancy and spherical designs. His lectures revisited a classical lower bound due to J. Beck on the inherent irregularity of any finite point set on a sphere, presented through entirely elementary techniques. Participants encountered the Stolarsky principle—an elegant identity linking uniformity to the sum of pairwise distances—and the linear programming method, a powerful tool for establishing tight bounds on highly symmetric structures such as spherical designs and codes.

Ujué Etayo from CUNEF Universidad followed with a course on optimal point distributions and spherical t-designs, centred on a landmark 2013 theorem by Bondarenko, Radchenko and Viazovska. That theorem resolved a longstanding conjecture by proving the existence of near-optimal spherical designs in all dimensions—a breakthrough later honoured when Viazovska received the 2022 Fields Medal for related work on sphere packing. Etayo's lectures traced the proof in full detail and explored how the underlying ideas extend to more general curved manifolds.

Bianca Gariboldi of the Università degli Studi di Bergamo completed the trio with a course on irregularities of distribution and the Cassels–Montgomery lemma, a central tool demonstrating that certain shapes—such as discs or spherical caps—can never be distributed with perfect uniformity. Starting from the classical torus, her lectures generalised the argument to Riemannian manifolds, revealing unexpected connections between number theory, geometry and Fourier analysis.

The school operated across two sites: BCAM's headquarters and the UPV/EHU's Department of Mathematics on the Bizkaia campus in Leioa. After the first two lectures days, participants then formed small research groups for guided collaborative work the following days. On Friday morning, each group presented its progress to the assembled school.

The summer school was organised by Javier Canto (EHU) (he/him), Daniel Eceizabarrena (BCAM) (he/him), Ioannis Parissis (EHU and Ikerbasque) (he/him), and Mateus Sousa (BCAM) (he/him).