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## First year Master courses in Bordeaux

You will find below brief description of the first year master courses (which are taught in French). 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, 6 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.

### Geometry (Semester 2, 9 ECTS).

The course is an introduction to various aspects of geometry: differential, algebraic, analytic, starting from a common object: Riemann surfaces.

### Number Theory (Semester 2, 9 ECTS).

This course develops the basic notions of Algebraic and Analytic Number Theory: Dedekind rings, The Ideal Class Group, the Dirichlet units, L-functions, etc.

### Spectral theory and Distributions (Semester 2, 9 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, etc. It also gives 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 and Statistics (Semester 2, 9 ECTS).

This is a systematic introduction to the probability theory, covering topics like the elementary probability to the cetral limit theorem, martigales, Markov chains, etc. It also 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 courses of the academic year 2014-15. The courses will be taught in English.