M1 - Fundamental Mathematics (MF)

In this course unit, we study classical groups (linear, unitary, orthogonal, symplectic) from their algebraic (reduction, conjugation classes, etc.), geometric (actions, exponential map), and topological perspectives.

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This unit expands on the classical theory of modules over a principal ring and the fundamentals 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 devoted to an introduction to EDPs, followed by their solution using well-known numerical methods such as the finite difference and finite volume methods. A section focused more on analysis—necessary for the introduction to the finite volume method—will be devoted to the analytical solution of scalar conservation laws. Four programming lab sessions will use simple examples to illustrate the scientific computing tools covered in class.

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

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This course unit covers the classical theory of Banach spaces and also serves as an introduction to spectral analysis.

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English tutorial course for students in the M1 Fundamental Mathematics program who are seeking professional proficiency in English.

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An introductory course on field theory, with Galois's 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 mapping, the fundamental group, and coverings.

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This course explores various aspects of effective calculus in algebra (linear algebra over Euclidean rings, resultants, etc.) and introduces the fundamentals of algebraic geometry.

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

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This course unit complements the differential geometry course in the M1 Fundamental Mathematics program. In this course, we apply the concepts from that course to the study of classical Lie groups.

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Each student works on a project supervised by a researcher or a faculty member who is also a researcher. Students write a thesis of approximately 30 pages, which they defend orally before a committee. Students may work in pairs.

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