Résultats de la recherche: 11

FMA_2F002_EP - Topology and Multivariate Calculus (2024-2025)

MAA202 is divided into two parts, the first one more theoretical than and setting the foundations of the second one. The theoretical part covers the fundamentals of topology of normed vector spaces and of topology in finite dimension. The second part, more computational, covers differentiation and integration in several (real) variables.

Analysis (MAA 202) builds on the concepts and techniques introduced in MAA102 Introduction to Analysis. In particular, students cover notions in topology.

Catégorie: Bachelor 2

FMA_2S007_EP - Series of Functions, Differential Equations (2024-2025)

Prerequisite: MAA202

Series of functions, differential equations (MAA207) builds upon the topology notions studied in Topology and multivariable calculus (MAA202) 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.

Catégorie: Bachelor 2

FMA_3F002_EP - Topology and Differential Calculus (2024-2025)

The first part of the course MAA302 is devoted to the theory of topological and metric spaces in an abstract setting, including the basic notions of continuity, completeness, compactness, and connectedness. We then shift our focus towards the space of continuous functions on a compact set, with the important theorems of Arzèla-Ascoli and Stone-Weierstrass, as well as towards Banach spaces, including the following fundamental results in functional analysis: the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. The final part of the course is devoted to differential calculus on Banach spaces, studying in particular the important results of the inverse function and implicit function theorems. If time permits we will conclude with an abstract theory of optimization, with and without constraints.

Prerequisite: Real analysis (MAA102); topology of normed vector spaces and multivariable calculus (MAA202)

Catégorie: Bachelor 3

FMA_3S001_EP - Measure and Integration (2024-2025)

Prerequisite: MAA202

MAA301 proposes an introduction to the modern theory of integration. The first part of this course is focused on the construction of the Lebesgue integral, an extension of the Riemann integral to a class of functions much larger than the set of Riemann-integrable functions. With the Lebesgue theory of integration, passing to the limit in integrals of sequences of functions is an easy task which rests on the  verification of a few essentially optimal assumptions. The end of the course offers an introduction to Lebesgue spaces and the Fourier transform, with applications to physics. The abstract theory of integration discussed at the beginning of this course provides the setting

MAA301 is devoted to the modern theory of integration. After first constructing the Lebesgue integral, and explaining how it improves the Riemann integral, a major part of the course will be devoted to discovering the power and ease of use of this tool.

Applications in probability theory will then be briefly described. The course will finally provide an introduction to Lebesgue spaces and the Fourier transform, in order to demonstrate the usefulness of the theory for applications in physics and economics.

used in probability theory and stochastic analysis.

Catégorie: Bachelor 3

FMA_3S006_EP - Differential Geometry (2024-2025)

This course provides an overview of the classical differential geometry of curves and surfaces.
More precisely, we will study the local theory of (regular, parametrized) curves (curvature, torsion), regular surfaces, and the local theory (first and second fundamental forms) and intrinsic geometry (Theorema Egregium and Gauss-Bonnet theorem) of the latter.

Weekly exercise sessions form an integral part of the course.
The instructor will provide lecture notes covering the material seen in class.

Prerequisites: some basic linear algebra, as seen is any undergraduate class (such as MAA101 and 201), and familiarity with multivariable calculus (differential of functions from ℝ^n to ℝ^2, as seen for example in MAA202). The most important notions will be briefly reviewed during the first lecture.

Catégorie: Bachelor 3

MAA202 - Topology and Multivariable Calculus (2023-2024)

MAA202 is divided into two parts, the first one more theoretical than and setting the foundations of the second one. The theoretical part covers the fundamentals of topology of normed vector spaces and of topology in finite dimension. The second part, more computational, covers differentiation and integration in several (real) variables.

Analysis (MAA 202) builds on the concepts and techniques introduced in Analysis 102. In particular, students cover notions in topology.

Catégorie: Bachelor 2

MAA202T - Topology and multivariable calculus - Tutorat (2023-2024)

Catégorie: Bachelor 2

MAA207 - Series of Functions, Differential Equations (2023-2024)

Prerequisite: MAA202

Series of functions, differential equations (MAA207) builds upon the topology notions studied in Topology and multivariable calculus (MAA202) 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.

Catégorie: Bachelor 2

MAA301 - Measure and Integration (2023-2024)

Prerequisite: MAA202

MAA301 proposes an introduction to the modern theory of integration. The first part of this course is focused on the construction of the Lebesgue integral, an extension of the Riemann integral to a class of functions much larger than the set of Riemann-integrable functions. With the Lebesgue theory of integration, passing to the limit in integrals of sequences of functions is an easy task which rests on the  verification of a few essentially optimal assumptions. The end of the course offers an introduction to Lebesgue spaces and the Fourier transform, with applications to physics. The abstract theory of integration discussed at the beginning of this course provides the setting

MAA301 is devoted to the modern theory of integration. After first constructing the Lebesgue integral, and explaining how it improves the Riemann integral, a major part of the course will be devoted to discovering the power and ease of use of this tool.

Applications in probability theory will then be briefly described. The course will finally provide an introduction to Lebesgue spaces and the Fourier transform, in order to demonstrate the usefulness of the theory for applications in physics and economics.

used in probability theory and stochastic analysis.

Catégorie: Ingénieur 1A / Bachelor 3

MAA302 - Topology and Differential Calculus (2023-2024)

The first part of the course MAA302 is devoted to the theory of topological and metric spaces in an abstract setting, including the basic notions of continuity, completeness, compactness, and connectedness. We then shift our focus towards the space of continuous functions on a compact set, with the important theorems of Arzèla-Ascoli and Stone-Weierstrass, as well as towards Banach spaces, including the following fundamental results in functional analysis: the uniform boundedness principle, the open mapping theorem, and the closed graph theorem. The final part of the course is devoted to differential calculus on Banach spaces, studying the important results of the inverse function and implicit function theorems, and concluding with an abstract theory of optimization, with and without constraints.
Catégorie: Ingénieur 1A / Bachelor 3

MAA306 - Differential Geometry (2023-2024)

This course provides an overview of the classical differential geometry of curves and surfaces.
More precisely, we will study the local theory of (regular, parametrized) curves (curvature, torsion), topological and regular surfaces, and the local theory (first and second fundamental forms) and intrinsic geometry (Theorema Egregium and Gauss-Bonnet theorem) of the latter.

Weekly exercise sessions form an integral part of the course.
The instructor will provide lecture notes covering the material seen in class.

Prerequisites: some basic linear algebra, as seen is any undergraduate class (such as MAA101 and 201), and familiarity with multivariable calculus (differential of functions from ℝ^n to ℝ^2, as seen for example in MAA202). The most important notions will be briefly reviewed during the first lecture.

Catégorie: Ingénieur 1A / Bachelor 3