The CRM hosts a new edition of the Barcelona Introduction to Mathematical Research summer school

From June 30 to July 25, 2025, the CRM is organising a new edition of the Barcelona Introduction to Mathematical Research (BIMR), a summer school hosted at the Universitat Autònoma de Barcelona. The programme brings together 30 undergraduate mathematics students to explore research through hands-on projects, introductory minicourses, and mentoring activities. With over 20 research topics offered, ranging from analysis and dynamical systems to geometry, cryptography, and mathematical modelling, BIMR provides early exposure to the research environment in an engaging and collaborative setting.

From June 30 to July 25, the Universitat Autònoma de Barcelona is hosting a new edition of the BIMR 2025 – Barcelona Introduction to Mathematical Research summer school. This initiative is promoted by the BGSMath, the CRM’s training unit. The programme offers 30 undergraduate mathematics students the opportunity to engage with mathematical research for the first time, working side by side with active researchers.

At the heart of BIMR are the research projects, assigned according to the preferences of both students and supervisors. This year’s edition features over 20 projects across a wide range of fields, including analysis, PDEs, dynamical systems, algebra, geometry, cryptography and mathematical modelling. Participants will work on topics such as Gamma-convergence, braid theory, Hausdorff measures, optimal control systems, Baker domains, and chaotic dynamical systems applied to cryptography.

Four minicourses to open the doors to research

During the first two weeks, students attend four introductory minicourses, designed to provide an accessible yet rigorous view of cutting-edge topics in mathematics:

Introduction to Stallings automata and the geometry of subgroups of free groups, taught by Enric Ventura and Jordi Delgado (UPC), explores a geometric and algorithmic approach to studying subgroups of free groups. Using intuitive graphical representations and avoiding overly formal language, the course covers key results such as the Nielsen–Schreier theorem, the Schreier index formula, and the Hanna–Neumann inequality.

Markov chains, mixing times and applications, with Guillem Perarnau (UPC – CRM), offers an introduction to discrete Markov chains with a special focus on the concept of mixing time. The course combines spectral theory with classical examples (such as random walks on graphs) and explores applications in statistical physics and theoretical computer science.

Singular integrals, Hausdorff measures and capacities, taught by Laura Prat (UAB), introduces fundamental tools in harmonic analysis and geometric measure theory. The course covers Calderón-Zygmund integrals, Hausdorff measures applied to fractal sets, and various notions of capacity with applications in the study of removable sets and partial differential equations.

Mathematical modelling of cancer as an ecological and population dynamics system, with Tomás Alarcón (CRM), presents an innovative perspective on cancer biology through the lens of mathematics. The course shows how ecological tools and dynamical systems theory can help understand tumour behaviour as evolving ecosystems.

Beyond mathematics

In addition to the academic programme, BIMR 2025 includes social and mentoring activities, such as a round table on academic careers, a public outreach talk, and weekly informal gatherings. For students living outside the metropolitan area of Barcelona, the CRM offers accommodation fellowships throughout July.

With this programme, BIMR aims to bring the research experience closer to a new generation of mathematicians and help spark scientific vocations early in their careers.

The programme is organised by Natàlia Castellana (Universitat Autònoma de Barcelona, CRM), Marc Masdéu (Universitat Autònoma de Barcelona, CRM), Xavier Ros-Oton (Universitat de Barcelona, ICREA, CRM) and Olli Saari (Universitat Politècnica de Catalunya, CRM), with the support of the BGSMath network.