The first part of the course 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.
Prerequisite: Real analysis (MAA102); topology of normed vector spaces and multivariable calculus (MAA202)
Le cours Algèbre et théorie de Galois (FMA_3F003_EP) présente les concepts et outils fondamentaux de la théorie des anneaux et de l’arithmétique, en accordant une place particulière aux anneaux de polynômes et à la théorie des corps. L’étude des extensions de corps conduit à la théorie de Galois, qui permet d’en comprendre la structure à travers les groupes de symétries qui leur sont associés. Les séances d’exercices illustrent et approfondissent ces notions à travers de nombreux exemples issus de l’arithmétique et de la théorie des polynômes.
Prérequis : FMA_1F001_EP, APM_1S003_EP, FMA_2F001_EP, FMA_2S006_EP.
The central object of the course are complex functions of the complex variable. We will study the notion of complex differentiability, also known as holomorphicity, and the main properties of holomorphic functions. In particular, we will see that
- the integral of a holomorphic function along closed curves is always 0
- holomorphic functions are infinitely many times differentiable
- holomorphic functions that coincide on any arbitrarily small disc in an open connected set coincide on the full set.
It will be clear pretty soon that complex-differentiability is a much stronger requirement than the usual differentiability with respect to real variables.
We will also study the different singularities a complex function of the complex variable might have, the famous residue formula and its applications.
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.
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.
