3 credits

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.

3 credits

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.

3 credits

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.

3 credits

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.

1 credits | Pre-requisite: FSC600 or SCF600

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

1 credits | Pre-requisite: MAT601

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

3 credits

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.

1 credits | Pre-requisite: MAT601

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.