M1 - Fundamental Mathematics (FM)

In this course we study classical groups (linear, unitary, orthogonal, symplectic), in their algebraic (reduction, conjugation classes...), geometric (actions, exponential application) and topological aspects.

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This UE develops the classical theory of modules on a principal ring, and the basics of the theory of representations of groups by concentrating on finite groups.

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Description : Partial Differential Equations (PDE) are nowadays an essential mathematical object for the study and understanding of physical or biological phenomena. Their great complexity often makes them impossible to solve analytically; hence the need to use numerical solution methods.

This course is dedicated to the introduction of PDEs, then to their solution using well-known numerical schemes such as finite difference and finite volume methods. A more analytical part, necessary for the introduction of finite volume methods, will be devoted to the analytical resolution of scalar conservation laws. Four programming exercises will illustrate the scientific computing tools seen in the course with simple examples.

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The construction of solutions of partial differential equations (PDEs) and the theoretical study of their qualitative behavior is essentially based on the use of analytical results and in particular of functional analysis. This course presents the first important tools for the solution of PDEs via analytical or geometrical points of view. These tools will be implemented in the study of some examples of PDEs representative of large classes of equations.

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

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English tutorial for students in the M1 Fundamental Math program who are aiming for professional autonomy in the English language.

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

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This course develops Cauchy theory for functions of one complex variable, and introduces the notions of conformal representation, fundamental group and coverings.

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

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An introductory course in differential geometry, focusing on the notions of subvarieties of ^Rn, vector fields and flows.

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This course complements the differential geometry course of M1 Fundamental Mathematics. We apply the notions of this course to the study of classical Lie groups.

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Each student works on a project supervised by a researcher or a teacher-researcher. Writing of a thesis of about thirty pages, defended orally before a jury. It is possible to work in pairs.

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