Analysis (MAA 207) builds upon the topology notions studied in Analysis (MAA 202) to allow for a more profound study of functions. Examining functions as limits of simpler ones (e.g. for approximation problems) is made possible in a rigorous manner thanks to topological ideas. This provides the possibility of using crucial tools in many scientific fields; the most striking one being Fourier series (first designed to solve the heat equation and now ubiquitous in science and, in a hidden manner, in daily life). The second part of the course deals with a wide array of differential equations, permitting students to better understand complex physical questions.
Course description: MAA201 continues the study of linear maps between vector spaces, started in MAA101. The goal is to obtain simple and efficient models for these applications up to suitable changes of coordinates. The concept of duality is initially introduced in the general context of mere vector spaces. Then, the focus is put on vector spaces enjoying a richer structure, namely prehilbert spaces, which is available in most applications (e.g. in solid mechanics or in quantum mechanics). The geometry of these spaces, as well as their important transformations (e.g. normal or unitary maps) is also discussed.
MAA203 covers a wide-range of important notions in probability theory and focuses in particular on discrete and continuous random variables with examples in modelling. A particular emphasis is put on how to perform and use computer simulations.
MAA204 is an introductory course in statistics, with complements in probability. Topics include displaying and describing data, writing a statistical model, introduction to statistical inference, confidence intervals, approximations with the Central Limit Theorem.
The purpose of MAA205 is to use computer science and programming to solve problems in discrete mathematics, and vice versa. Topics include: graphs and their matrices, combinatorics and generating functions, elementary probability, sorting algorithms. The course consists of lectures and practical labs in python.
Algebra (MAA 206) is a continuation of Algebra (MAA 201) and covers objects in bilinear algebra. These objects, mainly quadratic forms, have fundamental applications (e.g. in Number Theory and Mechanics), and also lead to the study of algebraic objects; for instance, some special groups of matrices, whose applications in mathematics and physics are fundamental, from Number Theory and geometry to the classification of particles.
MAA 208 covers the very important topic of numerical linear algebra. Starting with recalling linear algebra’s basic concepts (i.e. vectors, matrices, addition and multiplication), we quickly concentrate on methods for solving linear systems. Students study typical direct and iterative methods together with their practical implementation. This permits them to compare the methods in terms of complexity depending on the size of the problem to solve. The emphasis is put on the practical resolution of the problems and the theory that is required to understand the behavior of the methods considered. Subtle notions such as condition number, order of convergence, etc. are covered and explained. The course finishes with a project which is defended in-class during the last week of the semester. Students are evaluated based on this their project presentation, a report, and coursework.
Optimization concerns the minimization or maximization of an objective function. It often relies on the computation of the gradient of this function. MAA 209 covers several aspects of the classical methods that are used in such problems. For instance, the gradient methods (or steepest descent), the non-linear conjugate gradient methods will be seen. A particularly important topic concerns the Newton-Raphson method which extends the mono-dimensional Newton method to higher dimension. Comp. Mathematics B follows Comp. Mathematics A (MAA 208), since linear algebra methods are heavily used. Applications to the computation of the eigenelements of a matrix or to the resolution of non-linear systems of equations are also studied. As before, the course heavily uses practical sessions which are taken under consideration for the grading.