The fourth edition of the Hypatia Graduate Summer School took place from 16 to 19 June 2025 in Barcelona. Addressed to PhD students, the school combined two advanced courses in number theory with a public colloquium, a film screening, and networking activities such as a research speed dating session.

From 16 to 19 June 2025, the CRM and the Institut d’Estudis Catalans hosted the fourth edition of the Hypatia Graduate Summer School, a training programme addressed to PhD students in mathematics. Organised by BGSMath, the graduate training unit of the Centre de Recerca Matemàtica, the school offers an intensive introduction to selected topics, combining formal lectures with opportunities for discussion and exchange.

This year’s edition featured two advanced courses. The first, led by Henri Darmon (McGill University) and Victor Rotger (UPC–CRM), focused on rational points on elliptic curves and deformations of modular forms. The sessions revisited the construction of algebraic points on elliptic curves from the perspective of the Birch and Swinnerton-Dyer conjecture and the Gross–Zagier formula, with particular emphasis on settings beyond complex multiplication. The course introduced participants to recent advances in p-adic methods and modular generating series that offer new strategies for identifying rational points in more general arithmetic contexts.

The second course, taught by Özlem Imamoğlu (ETH Zurich) and Paloma Bengoechea (UB–CRM), explored periods of modular functions over closed geodesics. Starting from the classical theory of complex multiplication and the algebraicity of singular moduli, the lectures extended the discussion to values of modular functions at real quadratic irrationals, defined via integrals over geodesics on the modular surface. The course combined proven results with current conjectures, opening a window onto an emerging area of research at the intersection of number theory and geometry.

One of the highlights of the programme was the Hypatia Colloquium, held at the IEC. In his talk “Modularity and other aspects of the Langlands program: walking in Wiles’ footsteps”, Luis Dieulefait (UB–CRM) presented a panoramic view of the development of modularity results over the past three decades, from Wiles’ proof of Fermat’s Last Theorem to recent generalisations of Serre’s conjecture and base change theorems. The colloquium also introduced the Rayuela conjecture, a proposal aimed at unifying several threads in the Langlands programme.

On its first afternoon, the school offered a screening of Le théorème de Marguerite (dir. Anna Novion, 2023), a French film portraying the personal and academic challenges of a young mathematician. The screening was followed by an informal group discussion.

Beyond its academic programme, the school provides a setting for community building among early-career researchers. On the third day, participants took part in a research speed dating session, in which participants rotated through short conversations in pairs. The goal of the activity was to encourage interaction across institutional and disciplinary boundaries, offering PhD students the chance to expand their professional network in an informal setting.

With more than 40 participants from institutions across Europe and North America, the Hypatia Graduate Summer School continues to serve as a platform for doctoral training, combining specialised content with opportunities for exchange and collaboration.