BIMR 2026 - Barcelona Introduction to Mathematical Research

This new edition of the ‘Barcelona Introduction to Mathematical Research’ Summer Programme, under the BGSMath, aims to attract students in Mathematics (mainly in their 3rd or 4th year), from any University (in Barcelona or elsewhere) into the first steps in mathematical research in different disciplines.
The students will participate in the following activities:

1) Working on a research project in Mathematics during July. This will be done under the supervision of an Advisor/Tutor, and may consist of reading one or a few research papers (or book chapters), or working on a small open problem.
2) Attendance to the Minicourses on Introduction to Research that will take place during the weeks of June 29 – July 3 and July 6 – 10.
3) Attendance to the Round Table: ‘Academic career in mathematical research: what to do and when’ and a General Public talk (topic still to decide), that will take place during the afternoon in mid-July.
4) Participation in weekly social events (coffee breaks during the minicourses, and one social pizza evening during the first two weeks of the programme).

The school students will participate in four courses on Introduction to Research, given by:

TBP

TBP

Maria Alberich | Universitat Politècnica de Catalunya​

A Glimpse into the Calculus of Variations

The calculus of variations is a tool used to attack a wide range of minimization/maximization problems, for instance finding optimal shapes and curves under certain conditions. A celebrated classical example is for instance determining the optimal form of a slide connecting two given points such that a ball rolling down along the slide arrives in the shortest time: the so-called Brachistochrone problem. Another famous example is the isoperimetric problem, which asks the following question: among all simple closed curves in the plane of a given length which one encloses the biggest area?

When searching for the minima of a function, we have seen the utility of calculating first and second derivatives with respect to one or several variables, but how does one derive (once or twice) the “time of sliding down” or the “enclosed area” with respect to curves? This course will give an introduction to the calculus of variations and answer this question among others. The lectures will provide some theoretic foundations as well as several classical examples from physics and geometry. I wish to end the course by highlighting some famous problems, solved recently or unsolved.

Gyula Csató | Universitat de Barcelona – Centre de Recerca Matemàtica​

Category Theory — abstraction as a research tool

Knowing mathematics gives you an advantage in life, sometimes just by the capacity of abstract reasoning. Category theory is the field of mathematics concerned with general structures and properties that repeat themselves in all areas of mathematics.
It could be called the ‘mathematics of mathematics’. Things that look completely unrelated in distant areas of mathematics are revealed to be governed by the same universal properties.
Knowing category theory can give you an advantage in mathematical research, sometimes just by being able to separate out the general abstract aspects of a problem so as to be free to concentrate on the specific aspects. (And in some areas of mathematics, it is outright indispensable.)

This minicourse is a practical introduction to category theory.
Although we start from zero, we will get far enough to see some of the most important universal properties in action. By focusing on the category of finite sets, we’ll get to some interesting insights on very elementary mathematics that you thought you already understood. A secondary goal of the minicourse is to get to appreciate that even when we think we understand something, there is always a deeper level of understanding.

Here are three riddles, as teasers:

– What does the existential quantifier in logic have to do with the convex hull in geometry? or the abelianisation of a group?

– In which precise sense are addition and multiplication of numbers dual to each other?

– What does the number e have to do with the groupoid of finite sets?

Joachim Kock | Universitat Autònoma de Barcelona – Centre de Recerca Matemàtica​

Universality in Celestial Mechanics

In this course we want to address the question of how complex can the dynamics of the N-body problem. We will see that, in some sense it contains all possible dynamics, which gives an idea of the extreme complexity of the model. Since this is an introductory course, we will focus on the different mechanisms that give rise to such phenomena and provide a general overview of the involved techniques.

Pau Martín | Universitat Politècnica de Catalunya – Centre de Recerca Matemàtica​

The school students will participate in the following activities:

• Round Table: ‘Academic career in mathematical research: what to do and when’.
• Weekly social events: Daily coffee breaks will be provided throughout the minicourses, and one social pizza evening will be held each week during the first two weeks of July.

Research project applications will open: March 3, 2026.
Research project applications deadline: March 27, 2026.
Resolutions: April 15, 2026.

School registration will open: April 15, 2026.
School registration deadline: June 21, 2026.

Summer Programme: June 29th – July 24th 2026.

Olli Saari

Universitat Politècnica de Catalunya – CRM

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