Card-index course

The quest for the infinity in mathematics

La quête de l'infini en mathématiques

Responsible Faculty: Sciences au carré

Teacher(s): Dominique Arlettaz

Course Timetable (Weekly)

Course

Spring semester
2 hours per week
28 hours per semester
Weekly
Teaching language(s): French
Public: Yes
Credits: 2.00, 3.00

Objective

- Discover the concepts of infinity, limit, continuity and divisibility and some of their links,
- Be aware that these concepts occur in many mathematical domains,
- Compare the size of some infinite sets and define an infinite countable set,
- Notice that infinitesimal arguments are the basis of calculus,
- Understand the importance of the concept of series in the history of mathematics, in particuler in order to approximate certain numbers or functions and in order to solve some partial differential equations,
- Discover the first algebraic objects, the abelian groups, and some of their specificities,
- Explore the divisibility in abelian groups,
- Define the direct limit of a direct system of abelian groups,
- Be curious with very simple algebraic questions and find examples of statements which are not probable but true.

Content

The course contains five chapters :
- Elements of set theory and of number theory. The idea of infinity appears as soon as we try to count by using positive integers. However, it is a surprise to note that two sets may have the same size even if one is a part of the other, or to compare the sets of integers, of rational numbers and of real numbers, by using the concept of an infinite countable set.
- The basic concepts in calculus. The discovery of differential and integral calculus is based on the definition of a limit. This provides also nice strange examples of improper integrals.
- Series. The series played an important role in the evolution of differential and integral calculus. The goal is to define this tool and to explore its impact on many mathematical domains. For instance, the Taylor series provide nice approximations of certain numbers or functions and the Fourier series are used in order to solve some partial differential equations (in particular the heat equation).
- Abelian group theory. The first basic concept in algebra is the concept of an abelian group. It is interesting to be curious and to observe very simple abelian groups with strange divisibility properties.
- Direct limits in abelian group theory. The construction of the direct limit of a direct system of abelian groups produces new very large groups and one can check some non-probable properties which are effectively true.

Evaluation

2 written tests (60 minutes)

Bibliography

Jean-Michel Kantor et Loren Graham : Au nom de l'infini, une histoire vraie de mysticisme religieux et de création (Pour la science, 2010) Ernst Hairer and Gerhard Wanner : Analysis by its history (Springer, 1996) Franck Ayres and Philip A. Schmidt : Mathématiques de base, cours et problèmes (Série Schaum, McGraw Hill, 1993) Jean Combes : Suites et séries (PUF, 1982) Mohammed El Amrani : Suites et séries numériques, suites et séries de fonctions (Ellipses, 2011) Lennart Berggren, Jonathan Borwein, Peter Borwein : Pi : a source book (Springer, 1997) Derek J.S. Robinson : A course in the theory of groups (Springer, 1982) Joseph J. Rotman : An introduction to the theory of groups, fourth edition (Springer, 1995) László Fuchs : Infinite abelian groups, Volume I (Academic Press, 1970) Paul Moritz Cohn : Algebra I (John Wiley, 1974) Nathan Jacobson : Basic Algebra I, second edition (Freeman, 1985) Serge Lang : Algebra, 3rd edition (Springer, 2002)

Programme requirements

Only the basic highschool mathematical knowledge is required. This course does not depend on the first mathematical course of the program "Sciences au carré" which will take place during the Fall Term.

Additional information

https://moodle.unil.ch/course/view.php?id=18749