M1 - Fundamental Mathematics (MF)

In this EU, we study classical groups (linear, unitary, orthogonal, symplectic) in their algebraic aspects (reduction, conjugacy classes, etc.), geometric aspects (actions, exponential map), and topological aspects.

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This course develops the classical theory of modules over a principal ring and the foundations of group representation theory, focusing on finite groups.

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Description: Partial differential equations (PDEs) are now an essential mathematical tool for studying and understanding physical and biological phenomena. Their extreme complexity often makes them impossible to solve analytically, hence the need to use numerical solution methods.

This course is dedicated to introducing EDPs and then solving them using well-known numerical methods such as finite difference and finite volume methods. A more analytical section, necessary for introducing finite volume methods, will be devoted to the analytical solution of scalar conservation laws. Four programming labs will illustrate the scientific computing tools covered in class using simple examples.

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The construction of solutions to partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results, particularly functional analysis. This course presents important initial tools for solving PDEs from analytical or geometric perspectives. These tools will be applied in the study of several examples of PDEs representative of large classes of equations.

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This course develops the classical theory of Banach spaces and also provides an introduction to spectral analysis.

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English tutorial course for students in the M1 Fundamental Mathematics program who wish to achieve professional autonomy in English.

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Introductory course on field theory, with Galois correspondence theorem as the main result.

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This course develops Cauchy's theory for functions of a complex variable and introduces the concepts of conformal representation, fundamental group, and coverings.

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This course develops various aspects of effective calculus in algebra (linear algebra on Euclidean rings, resultants, etc.) and introduces the first elements of algebraic geometry.

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Introductory course in differential geometry, focusing on the concepts of submanifolds of^Rn, vector fields, and flow.

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This course complements the differential geometry course in the M1 Fundamental Mathematics program. It applies the concepts covered in that course to the study of classical Lie groups.

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Each student works on a project supervised by a researcher or teacher-researcher. Students write a thesis of approximately 30 pages, which they defend orally before a jury. It is possible to work in pairs.

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