Algebra, Geometry And Number Theory
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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.
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Admissions Requirements
Bachelor's degree in Mathematics or equivalent English proficiency Admissions Deadline: 31 January
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Nice
France
6000
France
- 2 years
- Full Time
- On Campus Learning
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