SUMMARY

Geometry is one of the oldest branches of Mathematics, still vigorously alive; it has evolved from the axiomatic approach of Euclid to the study of complex spaces, and it has several applications today. Moreover, Geometry is of great value for several other fields such as Physics (String Theory, Integrable Systems), Representation theory (Lie Theory, D−modules, Perverse Sheaves), Coding theory and Cryptography (Algebraic curves over finite fields), and to the society in general (education, architecture, engineering, medical technology).

Surprisingly enough, the subject is sometimes not covered at Chilean schools, and many of our high school teachers lack the knowledge and tools to teach Geometry. These facts are clearly reflected by the results of national and international tests taken by Chilean students of high school and teacher training programs, such as PISA or INICIA tests.

It is our belief that Geometry, in particular the line we follow, will be a great contribution in these directions. Through this Research Teams Grant Geometry at the Frontier, we look for a consolidation of our research team as a leading group on the subject. We will study research problems at the frontier of our field, and we will contribute to the national educational problem we stated before. Specifically we propose the following.

• To continue and expand our research in Geometry through the study of curves, surfaces, other related geometrical objects and their algebraic symmetries.

• To spread the knowledge of Geometry at school and university levels, incorporating the use of new technologies and through the activities proposed.

The Chilean Complex Geometry group has been important for the development of Mathematics in Chile. Its research production has proven to be fruitful, as shown by an important number of high‐impact publications, a wide network of collaborators, the organization of several international meetings and the fact that numerous former students of the group (over thirty) are now researchers and professionals in Universities all over the country and abroad. This number is substantial considering the universe of around three hundred active mathematicians in Chile.

This Research Team grant will give us the possibility to bring together researchers from the areas of complex and algebraic geometry. This grant and the critical mass provided by the group will enable us to overcome the difficult problem of isolation in Chile. We hope not only to learn techniques and theories from each other but also to join efforts to solve common problems. This grant will allow the group to have a deeper impact in the scientific international community than each of its individual members may have by him/herself, as it will allow us to work together in organizing intensive workshops in the country and abroad, thus bringing forward our research and broadening our network.

Since our researchers are widely distributed throughout the center and south of the country, we also expect a strong social impact in Chile for this project, far beyond the individual ones. In this grant we are proposing to create ready‐for‐ classroom‐use material in Geometry for students and teachers, that will prevail in time, and to organize workshops across the country to show them how to use it. We will also encourage young women and girls to go into Mathematics via dedicated activities: Math camps and a mentoring program.