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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. The ALGANT master course aims at introducing students into the latest developments of this fascinating subject.

These new developments 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.

Contents

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.

ALGANT consortium offers a two-year world-class integrated master course. 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). Algant in Milan comprises leading specialists in 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 A

* Fundamental; courses of this type are usually part of Bachelor programs

Type B* At least 60 credits

* 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).

Type C Around 12 credits

* Related courses, offered to ensure necessary variety in order to answer students' expectations and to broaden the scientific spectrum of the program. They appear on a list prepared by the teaching staff.

Type D At most 12** credits

* Free choice credits. This includes any Type A course taken during the two years of the ALGANT program.

Type E* At least 30 credits

* 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 (a formal convention is needed which describes the student's project). The project leads to the Master Thesis, which has to be defended. The research project can be split over the two years. A fair amount of individual work, which is one of the characteristic features of a master programme, will be required.

Type F Around 6 credits

* General training: language training, training on information technology or towards getting a better acquaintance with the world of entrepreneurship. Relational. (Note that although students can include some language training into the study program, language training is offered independently as part of the normal accompanying measures.)

*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. (In the above description, the word "about", can be replaced by an upper and a lower bound. The important thing being that the final sum is 120.) Most of the work on the research project will be performed in the second year, possibly as an internship in industry. So in the first year a student will generally go over the fundamentals of level 1 in the subject and acquire the credits in general training. The courses taught at level 1 on the four sites are highly compatible in the sense that they allow to follow the level 2 courses in any of the other sites. According to his/her abilities and aspirations the student will choose a particular direction/specialization, which will determine were he/she will spend the second year (recall that mobility is required).

Mobility scheme

The specialisation year 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. As a general rule, they spend their first year in one place. Then, students who have started with a third country partner will have to spend the second year in Europe, while students who have started with a European partner may not spend more than a semester with a third country partner. Students holding an Erasmus Mundus scholarship will of course have to abide to the rules set out in the Erasmus Mundus programme. More precisely, in ALGANT, such students from category A will have to spend all of their time in Europe and those from category B will have to spend their study period in at least two of the European partner countries. At least two of the contries visited must be different from the country in which the Erasmus Mundus scholarship holder has obtained his/her last university degree. In fact a number of Mundus scholarships for category B students, which will be determined every year, will be reserved for students willing to visit Stellenbosch/AIMS. The student might actually work on his/her research project at any institution setting up a convention through the consortium, provided that this work is performed under the (joint) supervision of a member of the fourth semester partner hosting institution. The mobility scheme will be reviewed every year, i.e. each academic year along with the review of the Student agreement.

Requirements

Funding

In ALGANT the amount of the full-study scholarships is variable. It is higher for Third-Country master students (Category A scholarships) than for European master students (Category B scholarships) and can be further reduced (to zero!) for those interested students that are selected by the ALGANT Commission, meeting the requirements of ALGANT excellence (Category C scholarships). More specifically:

* Category A Scholarship is an Erasmus Mundus Scholarship (possibly) provided by the European Community and can be awarded to masters students selected by the ALGANT Consortium who come from a country other than a Member State of the European Union, Iceland, Norway, Liechtenstein, Turkey, the Western Balkan countries or Switzerland and who are not residents nor have carried out their main activity (studies, work, etc.) for more than a total of 12 months over the last five years in such a country.

* Category B Scholarship is also an Erasmus Mundus Scholarship (possibly) provided by the European Community and can be awarded to any masters students selected by the ALGANT Commission who do not fulfil the Category A criteria defined above, for example, European students.

* Category C Scholarship is any other kind of Scholarship (possibly) provided by a very broad array of concerns such as math associations, research organizations, public institutions and private non-profit foundations (private self funding included).

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