This course is designed for graduate students working on their thesis. It gives them the opportunity to enhance their writing abilities and develop their critical thinking. It attempts to help students achieve greater competency in reading, writing, reflection, and discussion emphasizing the responsibilities of written inquiry and structured reasoning. Students are expected to investigate questions that are at issue for themselves and their audience and for which they do not already have answers. In other words, this course should help students write about what they have learned through their research rather than simply write an argument supporting one side of an issue or another.

This course is an introduction to the theory of groups which gives access to the many uses of theory of groups in mathematics. The central concepts are the structure and the actions of groups. Classifications of groups of small orders and of simple groups serve as motivation throughout the course. The themes addressed are: group actions, Sylow theorems, Semidirect Product, type finite Abelian groups, linear groups, projective groups and representations of finite groups. We derive two families of finite simple groups.

This course aims to teach students the mathematical maturity and the rigor of the methods of classification and resolution of partial differential equations, essential to mathematicians, engineers and computer scientists. It enables students to acquire the basic theoretical tools for manipulation of equations in partial derivatives frequently encountered in mathematical models applied to problems from mechanics, acoustics, hydrology, etc. Topics covered: total differential, integrating factors, linear equations of first order, partial differential equations in partial derivatives of order n to two variables, non­linear first­order partial differential equations, equation of heat and wave equation.

The main objective of this course is to initiate the students to the concept of random processes used in modeling of random phenomena. It focuses on the discrete Markov process or more commonly Markov chains. In the case of homogenous Markov chains we consider the set of States, the transition matrix, the initial distribution and the distributions at different times, the Chapman­Kolmogorov relationship, classification of States (stability, periodicity and recurrence), absorption in stable classes, stationary distribution, Newton diffusion gas problem, problem of the players ruin, one­dimensional random walk, multidimensional random walk, and study of the Poisson process and queues theory.

The objective of the course is to introduce students to scientific research. Topics covered are: interest and research objectives; methodologies used in scientific research, and how to define a problematic; data collection; documentary research; analyze the collected knowledge; structure of a Master thesis; write a report; write the bibliography; make a scientific poster; and how to approach making an oral presentation.

The first part of this course is designed to introduce students to the use mathematical language and reading and writing mathematical proofs as well as to the concepts and basic discrete math results which will be then used in the curriculum of mathematics and computer science. In the second part, we introduce some more advanced concepts in discrete mathematics with emphasis on mathematics used in algorithms (order, lattice, Boolean algebra, graph theory). It covers the following topics: mathematical methodology (logic, truth table); sets and relations; Boolean algebra; elements of arithmetic (Bézout's theorem, Fermat's little theorem, Euclid's algorithm); recursion and induction (linear and non­linear recurrence); graph theory (basic definitions, graph oriented and non­oriented, paths, matrices of incidence, Euler's theorem, planarity, coloring, labeled graphs, trees); mesh (introduction to partial order structures, terminal upper and lower); Boolean algebra (Boole lattice and rings of Boole, Boolean functions, normal forms); and rational and finite­state machine.

In mathematical analysis, a distribution (also called generalized function) is an object which generalizes the notion of function and measurement. The theory of distributions extends the notion of derivative to all locally integrable functions and beyond, and is used to formulate solutions to certain partial differential equations. They are important in physics and engineering where many discontinuous problems naturally lead to differential equations whose solutions are distributions rather than ordinary functions. The theory of distributions was formalized by the French mathematician Laurent Schwartz leading him to win the Fields Medal in 1950. Its introduction uses linear algebra and topology concepts centered around the idea of duality. We look for the origin of this theory in the symbolic calculation of Heaviside (1894) and in the introduction by physicists to the "Dirac function" (1926). The objective was to generalize the notion of function, in order to give a correct mathematical meaning to objects handled by physicists. It was necessary to keep the ability to do operations such as derivations, convolutions, and transformations of Fourier or Laplace. The distribution of Dirac is an interesting example of distribution because it is not a function, but can be represented informally by a degenerate function which would be void on its domain of definition, except 0 and the integral would be 1. In reality, quite strictly, it is the limit of a sequence of integral functions 1 distributions and converging uniformly to 0 on all compact does not contain 0. Such a mathematical object is useful in physical or signal processing, but no regular function has these properties.

Lie groups are groups equipped with a structure of manifold compatible with their group structure. Combining topology, algebra and geometry, they play a fundamental role in many branches of mathematics, but also in theoretical physics. This course is an introduction to the theory of Lie groups and Lie algebras, through the broad matrix Lie groups.

The objective of this course is to introduce students to the world of mathematical modeling and numerical simulation. The modeling and simulation have taken considerable importance in recent decades in all areas of science and industrial applications. Numerical analysis is the discipline that designs and analyzes methods or algorithms. Numerical simulation enables mathematicians to tackle problems far more complex and concrete than before, from immediate motivations that are industrial or scientific, which can provide answers to both qualitative but also quantitative questions; this is mathematical modeling. On the other hand, the scientist who was able to numerically simulate the problem does not stop there: he then wants to be able to change some parameters to improve or optimize the operation, performance, or the response of a system by maximizing (or minimizing) the associated functions. It is precisely the goal of optimization that provides theoretical or numerical tools to do this. Numerical analysis and optimization are therefore two essential and complementary mathematical modeling tools.

Topics selected from recent literature on mathematics are studied in depth. Students will participate in a series of conferences presented by experts.

Advanced Topics selected from recent literature on mathematics are studied in depth. Students will participate in a series of conferences presented by experts.

The spectral theory, an essential branch of functional analysis, applies to both pure and applied mathematics (differential equations or PDEs, theory of Von Neumann algebras) as in physics and chemistry. The purpose of spectral theory is, for some Endomorphisms of a hilbertien space, to obtain reduced shapes similar to the canonical forms of Jordan for the Endomorphisms of a finite­dimensional vector space and diagonal forms for the Endomorphisms of a hermitian space vector finite­dimensional hermitian. The theory of Hilbert­Schmidt applications was encountered for the first time in the integral equations, to build a first generalization of the results obtained in the finite­dimensional. In fact, the natural setting of this generalization is that of compact applications, studied by F. Riesz. Nevertheless, the case of the more general Endomorphisms escapes this framework; it is a subject for the spectral theory of Hilbert, which uses the techniques of integration. This course is intended primarily for students in the Master of mathematics, and it presents the mathematical tools of spectral theory: Basic Elements of functional analysis (normed spaces and spaces of Banach, spaces of Hilbert, continuous linear maps, duality, weak topologies), passage of the finite to the infinite dimension for continuous linear operators, dimension theory of compact operators, various forms of the spectral theorem, and self­adjoint operator theory.

Topics selected from recent literature on mathematics are studied in depth. Students will be responsible for presenting selected topics from the current scientific literature. They will be graded on relevance, critical analysis and presentation.

Master Thesis Report