Analysis

Analysis courses in SMA

(last modified 29.12.2022)

Contact person: Professor Maria Colombo

Bachelor (1st and 2nd years):

Bachelor (5th and 6th semesters): 

Master:

 

Description :

The track in Mathematical Analysis offers advanced knowledge of traditional fields of mathematical analysis such as the theory of real (and complex) functions, ordinary and partial differential equations, functional analysis, optimization. One of its focuses is the study of linear and nonlinear differential equations: these objects model problems from physics, chemistry, biology,  economy and studies the fundamental properties of their solutions such as existence, uniqueness, regularity, asymptotic behavior.

During the first and second bachelor years, the students are requested to follow the core courses "Analyse I - IV" as well as the course "Espaces métriques et topologiques" (BA5). Their contents will be at the basis of all subsequent courses on analysis.

In the third bachelor year, the students are strongly advised to follow  "Equations différentielles ordinaires" (BA5), which studies in particular local and global existence of solutions, asymptotic behavior, stability of fixed points and applications, such as dynamical systems or biology; "Analyse fonctionnelle I" (BA6), which presents fundamental concepts about normed linear spaces and linear operators; "Mesure et intégration" (BA5) to develop the modern concept of integration that lies at the basis of several studies such as the probability theory.

Finally, a big part of today's research in analysis has to do with partial differential equations; for this reason, an introductory class as "Introduction aux équations aux dérivées partielles" (BA5), sometimes complemented by an extra course on  "Espaces de Sobolev" (BA5) is of fundamental importance. These two courses develop the classical theory of linear/quasilinear elliptic PDEs, as well as the fundamental role played by Sobolev spaces in this theory.

Students wishing to specialize on geometry should take the course on differentiable manifolds "Introduction aux variétés différentiables" (BA5).

At the master level, there are courses on partial differential equations ("Linear PDEs", "Calcul des variations"), harmonic analysis and functional analysis.

There are also many master courses on related topics, for example a course on Riemannian geometry and a course on stochastic PDEs. Nowaydays an area of research, aiming at the simulation of many partial differential equations, is related to methods of scientific computing. It uses elements of mathematical analysis, numerical analysis and computer science. Interested students can look at more applied courses such as "Numerical approximations of PDEs".

The courses mentioned in this page may not be given every year and their name can vary slightly from year to year. The topics of some of the master courses may be accessible also to bachelor students with a specific interest and the students are invited to ask the professor of the course about this possibility. Almost all the courses mentioned above should be taken during the third year of your bachelors and during your masters by students who decide to focus on the Analysis track. Moreover, topic courses may be proposed and vary each year to introduce students to a particular area of current research. Students can consult as well the webpage of the research groups at EPFL, which may have informations on the suggested courses to be able to conduct research in the given group. Moreover, suggestions on important courses to take may come also from the project supervisor and may be related to the particular topic of the project itself.

Some other Math courses related to analysis: