APEX - MATHS

The program focuses exclusively on Number Theory and Topology. These two areas have the advantage of being full of beautiful and striking results, while remaining accessible through the study of numerous examples. They also happen to be related to one another in unexpected and deep ways, as we hope that the participants will discover!

More specifically, we will focus on the following notions:

1) Number theory:

• Euclidean division and the prime numbers
• Modular arithmetic (congruences, Chinese Remainder Theorem)
• Field extensions, algebraic numbers
• Algebraic integers and prime factorization in number fields
• Field automorphisms

2) Topology:

• Notions of metric spaces and topological spaces
• Isometries, continuous maps, and deformations
• Fundamental groups and universal covers
• Covering maps and topological surfaces

The program is designed for students who have a strong background in high school mathematics (basic algebra, polynomials, function theory, some familiarity with complex numbers, though not essential, is recommended).

Lectures will be delivered by the following university professors:

1. Andrei Negut (EPFL and IMAR)
   Andrei Negut obtained his PhD at Columbia University under the supervision of Fields medallist Andrei Okounkov. He is a specialist in geometric representation theory and in the theory of quantum groups. After having spent close to a decade at MIT he is now moving to the Ecole Polytechnique Fédérale de Lausanne.
2. Olivier Schiffmann (CNRS, Université Paris-Saclay and IMAR)
   Olivier Schiffmann's work is in the field of geometric representation theory and algebraic geometry. He has written over 50 research papers and has been invited professor in places such as Japan, China, Italy and USA. He is now a senior researcher (Directeur de Recherches) at CNRS and affiliated with IMAR.
3. Anne Moreau (Université Paris-Saclay)
   Anne Moreau is an expert in algebraic and geometric representation theory; her contributions span a broad array of topics ranging from invariant theory to motivic integration, vertex algebras and mathematical physics. She has organized semester-long thematic programs in Brazil and Canada and has traveled around the world to give talks in numerous conferences and summer schools. She joined the faculty of the Mathematics Department of Paris-Saclay University in 2020.
4. Emanuele Macri (Université Paris-Saclay)
   The work of Emanuele Macri lies in the area of algebraic and enumerative geometry. He is a recognized expert in birational geometry and in the study of moduli spaces of bundles on threefolds. He has traveled to present his research in numerous conferences around the globe. In 2023 he was awarded the Medaglia dei XL per la Matematica by the Italian National Academy of Science. Before joining the Mathematics Department of Paris-Saclay University in 2019 he was a professor at Northeastern University.

List of Mentors includes:

1. Bogdan Carasca (PhD Student, Berlin University)
2. Clarence Kineider (PhD Student, Strasbourg University)
3. Paul Laubie (PhD Student, Strasbourg University)
4. Mihai Pavel (Post-Doc, IMAR)
5. Robin Riegel (PhD Student, Strasbourg University)
6. Thomas Saigre (PhD Student, Strasbourg University)
7. Calin Spiridon (PhD Student, IMAR)
8. Andrei Stoenica (Phd Student, IMAR)
9. Romeo Troubat (PhD Student, Strasbourg University)
10. Alexandra Melnig (Young Researcher, UAIC & IMOM)

Scientific research differs from most learning activities in one essential way: before you can try and answer a question, you must formulate it! In fact, finding the right question to ask is quite often the hardest and most crucial part of the research process. It can only be achieved through a deep analysis of numerous examples - and usually comes after much trial and error!

At APEX, participants are led to explore a mathematical theme, as in an actual research process. The focus is on investigation!

Through daily courses (around two hours in the morning, from 9 am to 11 am) given by distinguished university professors and problem sets which are handed out at the end of each lecture, the participants are encouraged and guided to think deeply and in new ways about some natural mathematical notions (such as numbers or shapes), to discover some pattern or phenomenon, make conjectures, and develop the right set of tools to prove -or disprove!- these conjectures. As in real research, time is not an issue. Exchanging your ideas and sharing your intuition with others is - in addition to being fun! - a crucial part of the exploration process.

Students are assigned to one PhD student, who serves as a mentor. One PhD student would typically be mentoring 4 to 5 students and lead daily informal small investigation groups through the discovery process.

Students are expected to attend lectures and seminars and spend a good portion of the rest of the day working on the problem sets, alone or collaboratively. Excellent facilities (rooms with boards for small groups) will be available to facilitate interaction among participants.

All the lectures and seminars are presented in English.

A typical weekday runs as follows:

• 2 hours of lectures in the morning, delivered by one of the four professors, followed by
• 1 hour of working group (typically 4-6 students)/seminar, led by the Mentors, where the main investigation directions about the theme of the day are decided.

After lunch, the students are encouraged to work on their own or in groups supervised by the PhD students on the problem sets. Permanent interaction with the distinguished professors is encouraged and will be possible all day long.

• Outdoor activities will typically be planned for the late afternoon, before dinner
• in the evening, on some weekdays, to be decided ad-hoc: 1 hour of seminar in larger groups.

Some independent stand-alone lectures on various topics in maths (given by some of the mentors or professors) may be organized.

During the weekend, outdoor activities will be offered to the participants (such as sports, hiking, tourism (see detailed program here) In particular, a weekend in Coronini, to visit the area along the Danube, a visit to the historical sites of the city Timişoara (during the weekend of your arrival in Romania), as well as daytrips to the spectacular Reşiţa surroundings will be organized.