## General presentation

Traditionally number theory used the methods of algebra and analysis, to solve problems such as finding the number of integral of solutions of equations. In recent times geometric methods have been playing a more important role. Also, number theory has important applications in areas such as cryptography, theoretical computer science, and numerical mathematics. These led to a unification of number theory. The ALGANT course aims at introducing students into the latest developments of this fascinating subject.

## Main focus on ALgebra, Geometry And Number Theory.

The Master Course is built on wider Master programs in mathematics and our course allows for the choice of optional courses in other areas of mathematics, physics, computer science, and the history and philosophy of science. Still the main focus is on Algebra, Geometry and Number Theory because:

- these are subjects that have a much greater tradition in Europe than anywhere else in the world;
- some of the most important recent advances in mathematics have taken place in our field and our staff is perfectly in synch with these advances: our departments have a worldwide reputation in these fields;
- Number Theory is a "royal way" into higher, contemporary mathematics, thus permitting to attract good students who might not have had the necessary formal training at the bachelor level; starting with classical, "concrete" problems, such students will be quickly introduced to the more sophisticated algebraic and geometric techniques that lie at the foundations of so much current work;
- as is well-known, but still unexpected few years ago, the most recent advances in Algebraic Geometry have led to important applications (cryptography, error correcting codes, etc.); we offer training towards employment in areas were expertise in these fields is necessary for the development of applications.

## Study program.

ALGANT consists mainly of advanced courses in the field and of a research project or internship (leading to a master thesis). More precisely courses are offered in: algebra, algebraic geometry, algebraic and geometric topology, algebraic and analytic number theory, coding theory, combinatorics, complex function theory, cryptology, elliptic curves, manifolds. Students are encouraged to participate actively in seminars.

The partner departments offer a compatible basic preparation in the first year (level 1), which then leads to a complementary offer for the more specialized courses in the second year (level 2). Overall the specialized courses cover a very wide spectrum of subjects.

Geometers study geometric properties of sets of solutions of systems of equations. In algebraic geometry the equations are given by polynomials. Number theorists consider so-called Diophantine equations, that is, systems of equations that are to be solved in integers. Traditionally, the methods of number theory are taken from several other branches of mathematics, including algebra and analysis, but in recent times geometric methods have been playing a role of increasing importance. It has also been discovered that number theory has important applications in more applied areas, such as cryptography, theoretical computer science, dynamical systems theory and numerical mathematics. These new developments stimulated the design, analysis and use of algorithms, now called computational number theory. They led to a unification rather than diversification of number theory. For example, the applications in cryptography are strongly connected to algebraic geometry and computational number theory; algebraic number theory, which used to stand on its own, is now pervading virtually all of number theory; classical objects like zeta functions, introduced with the analytic approach to number theory, have been generalized to become effective tools encoding the number of solutions of Diophantine equations. They have been given a cohomological interpretation, and their study relies heavily on the study of the representations of Galois groups.

These developments have led to the theory of motives. Some of the most striking results obtained in the field are the proof of Weil's conjectures (Dwork, Grothendieck, Deligne), Faltings' proof of Mordell's conjecture, Fontaine's p-adic Hodge theory, Wiles' proof of Fermat's Last Theorem and Lafforgue's result on Langland's Conjecture. As suggested above, great progress has also been achieved in primality tests and factorization methods, and the development of efficient computer algorithms. Again we are pleased to note that the very recent approach to p-adic rigid cohomology by Kedlaya has led to better algorithms for the computation of points on algebraic varieties. It should be emphasized that our departments have actively contributed to the above developments: a forerunner of Fontaine's theory has been developed by Barsotti in Padova, where Dwork also has taught; Murre, in Leiden, has made substantial contributions to the theory of motives; the most used computer program in the world for computational number theory has been developed by a group of researchers under the supervision of Cohen (Bordeaux; PARI/GP), and one of the better known algorithms in the field bears the name of Lenstra (Leiden). Edixhoven (Leiden) is a well-known expert in the theory of modular varieties, whose study is crucial for Langlands' programme. Darmon (Montreal) is one of the better-known experts on the field that has developed following Wiles' proof of Fermat's Last Theorem. The Padova department has participated in all the important developments related to p-adic cohomology, which these days allow new effective approaches to important classical results. (More information on the staff's scientific achievements and the complementarity of their expertise will appear from examining the attached summary curricula.)

**Academic quality.** The partner institutes aim to combine excellence both in research and education. They attract many visitors from abroad, organize workshops and give specialised courses for students. Some have a very long and well established tradition. One could recall that Galilei has taught 18 years in Padova (1592-1610), that Descartes has published his *Discours de la Méthode* in Leiden (1637) or that Hadamard was in Bordeaux when he proved the Prime Number Theorem (1896), but, more to the point, the teaching staff currently applying for the recognition of this master programme has an excellent track record over the past quarter of a century. Paris Sud stands out as it counts among its faculty in mathematics three recipients of the Fields Medal. Let us indicate that a number of world-class textbooks are based on courses of this programme (see the end of this paragraph).

Many members of staff have held positions at internationally recognised institutions before joining their current departments, so for instance the main coordinator held a junior position at Harvard and the contact person in Padova held one at Princeton. Some are world leaders in their field (see the attached curriculum vitae for details). Also, the academic staff has already repeatedly collaborated at the personal and at the institutional level. For research: Bordeaux and Leiden are part of the RT network Galois Theory and Explicit Methods-GTEM and have set-up a bilateral International Scientific Cooperation Programme-PICS; Bordeaux, Paris Sud, Leiden and Padova were part of the RT network Arithmetic Algebraic Geometry-AAG coordinated in Milano; researchers in Montreal collaborate on a regular basis with colleagues in Milano and Padova; CMI has relations with Milano and Paris; researchers in Stellenbosch collaborate with colleagues in Bordeaux. Paris Sud is one of the six institutions supporting AIMS.

The students also profit from interacting with the many foreign visiting scholars who visit our departments on a regular basis. Scholars commit themselves to circulate within the consortium and to the benefice of the largest part possible of the students enrolled in the programme.

The scientific environment and facilities we offer to ALGANT students are exceptional. All partner universities are research intensive universities. We have sufficient staff to guide the students individually. Our departments run lively weekly seminars, for instance the Number Theory Seminar in Bordeaux has led to the foundation of the internationally recognised Journal de Théorie des Nombres de Bordeaux and the Intercity Number Theory Seminar allows to attract to Leiden the best experts in the field. This makes for a very rich student life, open to the most recent advances in research. Bordeaux organizes a unique series of lectures by leading experts on current aspects of research in mathematics and disposes of facilities reserved for the students in the Master, Padova and Milano Mathematics Departments count among those with the most international relations in Italy. Montreal counts one of the largest and active research communities in our field in Northern America. CMI has just been reconstructed and attracts some of the best Indian students in mathematics. Stellenbosch is ranked number 2 among African universities and AIMS is a unique institution hosting a pan-African selection of students, preparing them for programme of international level.

## Formal credit requirements

For each student a program will be tailored individually to fit the student's previous curriculum (she/he might need complements), wishes and language proficiency (for adaptation, students have to be well to perform well). The programme will be included in the *Individual student
agreement*. However, to complete their degree students will have to obtain a minimum number of credits in specified kinds of courses. Moreover, it is expected that most students will follow one of the suggested predefined programs, actively participate in seminars and attend the joint intensive courses. Courses are taught at a specified level. In general courses of the first (resp. second) year will be of level one (resp. two).

Here is a list of various types of courses or learning activities, and for each the amount of credits required.

Type | Description | NB. of credits |
---|---|---|

Type A | Fundamental; courses of this type are usually part of Bachelor programs | N/A |

Type B* | Core and advanced courses from algebra, algebraic and analytic number theory and algebraic and differential geometry. Courses at the same level in the same field should not be followed more than once (e.g. the consortium will not accept a cursus studiorum in which appear for instance, two introductory courses to algebraic number theory). | At least 60 |

Type C | Related courses, offered to ensure necessary variety in order to answer students' expectations and to broaden the scientific spectrum of the program. | Around 12 |

Type D | Free choice credits. This includes any Type A course taken during the two years of the ALGANT program. | At most 12** |

Type E* | Research project, prepared under the supervision of an advisor from the teaching staff, which might be prepared at one of the consortium's departments, in another university or in any qualified partner research unit. The project leads to the Master Thesis, which has to be defended. A fair amount of individual work, which is one of the characteristic features of a master programme, will be required. | At least 30 |

Type F | General training: language training, training on information technology or towards getting a better acquaintance with the world of entrepreneurship. Relational. | Around 6 |

*Active participation in a seminar can be accounted for in Types B or E.

**6 of these credits can be moved to Type F.

Usually a student will acquire the 120 credits by adding the credits of types B, D and E to obtain about 100 and then modulate with credits of types C and F to attain 120. Students discuss their specific programme with their tutor/supervisor. Most of the work on the research project will be performed in the second year. In the first year a student will generally follow fundamental courses which are similar at all ALGANT universities. According to his/her abilities and aspirations the student will choose a particular direction/specialization in the second year.

**Mobility scheme.** The specialisation determines the mobility arrangement for a given student. Indeed, the consortium now completely covers the spectrum of specialisations in our field, thus offering a unique opportunity for students who want to be trained towards research in the area. For ALGANT, students can start the course at any of the partners.

## Acquired competencies and learning outcomes of the ALGANT Master Course.

The students having successfully completed the requirements of the ALGANT program will be well armed to pursue a career in research by preparing a Ph.D. or by directly applying for a job in the many companies that are looking for the know-how we teach. They will be awarded a double degree composed of two nationally recognized degrees issued by the two hosting institutions, completed by a diploma supplement.

The degree obtained is aimed at getting research skills founded on a firm theoretical basis, and the communicative skills necessary to function on the international scene and in teaching. The successful student will be able to apply to high-level Ph.D. programs, but he can also envisage a career as a mathematical researcher outside universities. In particular, an appropriate choice of courses will allow the students to contribute to the applications in fields using technology based on cryptography and coding theory (network security, electronic chips, multimedia and mobile communication industry, etc.). More generally the competencies and learning outcomes acquired with this Master Course are:

- a good general culture in mathematics and computer science allowing to tackle very varied kinds of problems in different situations;
- the competence to use a scientific approach to problem solving, be it in fundamental research or for applications;
- the ability to gather and organize relevant information and to critically scrutinize the result of ones own work;
- proficiency in English and acquaintance with at least two European languages and cultures; general communicative skills;
- being able to work in a group as well as autonomously, assuming responsibilities of projects and (small) structures in industry, finance or administration.

It is worthy to note here, that a survey performed in the context of the "Tuning Educational Structures" project, sponsored by the European Commission, shows a positive image of mathematics graduates with employers, so that a large majority of them finds a job at their level of qualification. This will be even more true for students graduating from an excellence program such as ours.