Algebra (MAA101) is a fast-paced course which provides students with an overview of the most useful techniques of linear algebra.

Analysis (MAA102) is an introductory-level mathematical analysis course that provides a well-balanced approach between calculus and foundational notions; it is designed to equip students with the fundamental analytical tools required in all scientific fields. In particular, this course covers derivatives and function approximation in one real variable. It also introduces students to important mathematical concepts which will be expanded upon later in the program; namely, the basics of topology on the real line.

Discrete Mathematics MAA 103 (Year 1) has two main objectives: (i) teach fundamental concepts in discrete mathematics, which are the building blocks of many different areas of science and of advanced mathematics (ii) teach how to write proofs. The cours strarts with elementary logic (e.g. quantifiers, different methods of proof), sets, and functions. The second part of the course introduces students to combinatorics and probability (on finite sets).

The lectures will closely follow the textbook Mathematics: A Discrete Introduction (3rd Edition) by Scheinerman. For a different presentation and broader applications concerning computer science, one may have a look at Discrete Mathematics with Applications by Epp.

Course webpage: http://www.cmap.polytechnique.fr/~kortchemski/dmaths/




Discrete Mathematics (MAA103) begins by introducing students to the central notions needed to pursue advanced mathematics, such as elementary logic (e.g. quantifiers, different methods of proof), sets, and functions. The second part of the course introduces students to combinatorics and probability (on finite sets). Course material is supplemented with examples and applications, such as graphical modeling and generating functions.
Algebra (MAA104)  introduces students to more conceptual algebraic subjects. More precisely, students explore the fundamental structures of algebra including groups, rings, and fields. Topics covered in this course are designed to prepare students for later questions related to symmetry (including those arising in physics) and number theory. This course also covers the study of polynomials, including their application, to further develop techniques acquired from linear algebra.
The first part of this course is focussed on the notion of Riemann integral. After introducing the notion of Riemann integrable function, we briefly discuss the basic properties of such functions. Next we present the classical methods for computing integrals (integration by parts, integration by substitution, integration of rational fractions, elementary abelian integrals…) Finally, we study the main notions and results on improper integrals.
The second part is dedicated to the study of ordinary differential equations, mainly first order linear differential equations and linear systems of ODEs, with a special focus on linear differential equations with constant coefficients.
The third part is devoted to the study of plane parametric curves. We will see their fundamental properties and how to sketch a plane curve. We will cover parametric curves in Cartesian coordinates and in polar coordinates.

 




The aim of Introduction to Numerical Analysis (MAA 106) is to provide the students with practical knowledge of basic mathematical algorithms and computer programming.

The course is divided into 5 cycles and covers several notions such as the representation of numbers, the complexity of algorithms, interpolation of functions, numerical integration, solving equations and optimization problems. We focus on the practical implementation of the methods, using in particular Python notebooks which enable to test easily the algorithms with multiple parameters for instance.

The last three weeks of the course are devoted to a more personal project that will allow the student to understand deeper a notion seen in class. A public defense of this work concludes the course. The grading of the course includes the project and a homework that needs to be given back at the end of each cycle of the course.