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First year Master courses in Bordeaux
You will find below brief description of the first year master courses. A more detailed description can be found
here (in French).
Modules and quadratic spaces (Semester 1, 9 ECTS).
The course will present the basic algebraic and analytic theory of quadratic forms, bilinear forms and symplectic forms, in particular the classification of these objects over the
fields of real and complex numbers, and over finite fields. Geometric applications to quadrics and conics will also be described.
Group Theory (Semester 1, 6 ECTS).
This basic introduction to group theory introduces the fundamental notions and results (group actions, cosets, Sylow subgroups, solvable and nilpotent groups)
of group theory. Linear groups over the real and complex numbers are studied, as well as over finite fields. The course ends with an introduction to Galois theory.
Complex Analysis (Semester 1, 9 ECTS).
This course starts from the theory of differential forms in the plane, with Stokes formula, to introduce the notions of holomorphic and harmonic functions. The
fundamental results are proved (Cauchy's Theorems, the residue formula, etc) and some of the applications of complex analysis are considered.
Functional analysis (Semester 1, 6 ECTS).
This first course in functional analysis introduces the fundamental notions of the subject. It describes the basic tools (Hahn-Banach and Banach-Steinhaus
theorems, the closed graph theorem, etc) and develops the duality of topological vector spaces. Examples are given and the theory of operators between topological vector spaces is
introduced.
Differential Geometry (Semester 2, 6 ECTS).
The course is an introduction to various aspects of differential geometry. It covers differentiable manifolds, fibre bundles, coverings, differential forms and the Stokes
formula. In addition, short introductions to Lie groups and homogeneous spaces on the one hand, and to Riemannian geometry are given, including the Gauss-Bonnet formula.
Algebraic Geometry (Semester 2, 6 ECTS).
The course is an introduction to fundamental notions of algebraic geometry. It starts with foundational matters: algebraic sets, Zariski topology, affine and projective
spaces, the Nullstellensatz. It then continues with the concepts of algebraic varieties, morphisms of varieties, irreducible components,
Krull dimension, smoothness. The case of curves is studied as an example, and as applications the course discusses the intersection of
curves in the projective plane (Bezout's Theorem), differential forms and the statement of the Riemann-Roch theorem and the Riemann-Hurwitz
formula.
Algebraic Number Theory (Semester 2, 6 ECTS).
This course develops the fundamental notions of algebraic number theory. It starts with preliminaries concerning commutative algebra
and then develops the basic theory of algebraic integers, rings of integers in number fields (norm, trace and discriminants,...) before
proving the basic results: unique factorization in ideals in Dedekind rings, finiteness of the class group, Dirichlet's unit theorem.
The examples of quadratic fields, cubic and biquadratic extensions and of cyclotomic fields are presented.
Analytic Number Theory (Semester 2, 6 ECTS).
This course develops some basic results of analytic number theory. It presents the definition and properties of the standard
arithmetic functions and of the Dirichlet convolution, presenting the link with Dirichlet series. The order of magnitude of
arithmetic functions and of averages of arithmetic functions is presented with the statement of the Prime Number Theorem and of the
Riemann Hypothesis, and the proof of Dirichlet's theorem on primes in arithmetic progressions. Additional topics are presented at the end
depending on time (e.g., transcendental numbers, sieve methods, partitions and generating series, exponential sums, equidistribution).
Spectral theory (Semester 2, 6 ECTS).
This course presents the fundamental results of spectral theory, with notions of Banach algebras, spectrum and spectral radius;
operators on Banach spaces and on Hilbert spaces, compact and normal operators, including the Fredholm alternative, Hilbert-Schmidt
operators and Sturm-Liouville equations.
Distributions (Semester 2, 6 ECTS).
This course is an introduction to the theory of distributions. After generalities concerning test functions, and the definition of
distributions, examples are given together with the basic operations on distributions. The study of the Fourier transform in the framework
of distribution follows. Applications to fundamental solutions of Partial Differential Equations (heat equation, Cauchy-Riemann
equation, Laplace operator) are given, and also an introduction to Sobolev spaces.
Probability (Semester 2, 6 ECTS).
This is a systematic introduction to the probability theory, covering topics from the elementary probability to the cetral limit theorem.
Probability and Statistics (Semester 2, 6 ECTS).
This course covers more advanced notions of probability (martigales, Markov chains, etc.), and includes a systematic introduction into the modern mathematical statistics (estimates, Neuman-Pirson theory, etc.)
Cryptology (Semester 2, 6 ECTS).
This course is an introduction to the modern methods and applications of cryptology. It discusses both symmetric and asymmetric cryptology,
and describes the standard algorithms and techniques in the subject (one-way functions, discrete logarithm problem, RSA, ...)
Algebra and formal computation (Semester 2, 6 ECTS).
The subject of this course is to introduce and study the main algorithms in use in algebra and arithmetic, and their implementations on computers. Those
algorithms concern the fundamental operations, sorting, linear algebra over a field or over the integers, polynomials, etc. Algorithmic complexity is
introduced. The programming part is done using the Computer Algebra system MAPLE.
Second year Master courses in Bordeaux
These courses are renewed each academic year.
Here (pdf, 118 KB) you will find the summaries (in English and French) for the year 2009-10 courses. The courses will be taught in English.
