The adventure of integers & The quest for infinity in mathematics

Semester: Autumn 2020
Schedule: Friday 8:30am à 10:00am
Classroom: 
Number of hours: 28

Modality of the course: Pre-sessional teaching

Objective

• Understand the first - and simplest - tool essential to any scientific endeavour: the concept of the whole number, which has made it possible to count, number and classify since the dawn of humanity
• To appreciate the breadth and diversity of mathematics and the countless links between the different areas of mathematics
• Exercise your curiosity with problems that can be formulated in a very simple way but whose solution may turn out to be extremely complicated, or even unknown to date
• Cover some basic mathematical concepts and understand simple but rigorous mathematical reasoning, which enables some fundamental mathematical theories to be demonstrated
• To demonstrate the fact that major advances in mathematics are always the result of collaboration between several researchers and involve many mathematical concepts
• Following the historical development of one of the great mathematical problems, the solution of the Fermat Conjecture, which kept the world of mathematicians on tenterhooks for more than three centuries, until it was abandoned in 1995.

Contents

The course will consist of four chapters:

• Prime numbers: the first chapter introduces the sets of numbers, the definition of prime numbers, the main properties of prime numbers, and an application to the theory of coding.
• Fermat's last golden theory - part one :  the second chapter introduces Pythagorean triangles, exposes the noncute; of Fermat's last théorème (1637), describes the first steps in its deémonstration and fully explains the proof of the case of the exponent n=4.
• Complex numbers: the third chapter presents the definition of complex numbers and the Gaussian plane, the solution of certain complex equations, the Gaussian integers and the complex expander function.
• Le dernier théorème de Fermat - seconde partie: the last chapter presents the history of the solution of Fermat's last théorème (énoncé in 1637 by Pierre de Fermat and déshown; in 1995 by Andrew Wiles) and allows us to follow - as in a detective novel - the countless efforts that have been made by scientists to solve this mystery.

Required

None, apart from the basic knowledge of mathematics corresponding to the level of the maturit; gymnasium.
This course can be taken as a complement to, and independently of, the second maths lesson given as part of the Science programme in the spring semester.

Mode of assessment

• Continuous assessment in the form of two knowledge tests (60 minutes)
• A seminar paper reporting on the reading of a chapter of a book related to the subject of the course or any other subject agreed between the student and the teacher.

NB: The corresponding credits cannot be validated for FGSE students (autumn and spring).

Semester: Spring 2021
Schedule: Friday 10:15am à 12:00pm
Classroom:  
Number of hours: 28

Course mode: Distance learning

Objective

• To become familiar with the notions of infinity, limits, continuity and divisibility, and to understand some of their relationships,
• Be aware that these concepts appear in many areas of mathematics,
• Compare the size of some infinite sets and cover the notion of dénombable infinity,
• Confirm that infinite reasoning is the basis of differential and integral calculus,
• Understand the importance that the concept of sine (i.e. the limit of a sequence of sums comprising ever more terms) has played in the development of mathematics;development of mathematics, from its origins to the present day, in particular to approximate certain functions or to solve certain partial differential equations,
• To cover the simplest algebraic objects that are abbreviated groups and some of their particularities,
• Exploring the notion of divisibility in abélink groups,
• Defining the notion of (direct) limit in group theory abéliens,
• Exercise your curiosity with simple algebraic notions that allow you to illustrate very surprising problems; in particular cover examples of assertions that are implausible but true.

Content

• Elements of set theory and number theory. The idea of infinity appears as soon as we count using the set of positive integers, but the first surprises come when we think of two sets of numbers that have the same size, but whose numbers are different;same size, but one of which is a part of the other, or when we compare the sets of integers, rational numbers and real numbers, in the light of the notion of an infinite denominable set.
• The foundations of differential and intégral calculus. To learn about infinity in mathematics, you need a thorough understanding of the notion of the limit of a function (or sequence), which is at the heart of the coverage of differential and integral calculus. This makes it possible to deal with étonnant examples of improper intégrales.
• The series. Developments in differential and intégral calculus have been built on the notion of the infinitely small and the concept of séries. The aim is to define the series and to show how they have become an extremely powerful tool for exploring many other mathematical concepts and for solving a number of major scientific problems. This is illustrated by Taylor's sequences, which provide excellent approximations to numbers or functions, and by Fourier's sequences, whose usefulness is illustrated by the solution to the heat equation.
• The theory of abbreviated groups. The simplest algebraic concept is that of an abbreviated group. The aim is to cover the basics of this theory and to exercise one's curiosity by observing very simple abacute;linkage groups that have similar properties in terms of divisibility.
• The direct limit in abacute group theory. The construction of the direct limit of a direct system of abéliens groups allows the construction of new (very large) abéliens groups and the discovery of some really surprising phénomes.

Required

None, apart from the basic knowledge of mathematics corresponding to the level of the matura gymnasium.
This course can be taken as a complement to, and independently of, the mathematics course;This course can be followed independently of the maths course given as part of the autumn term "Sciences au carré" "L'aventure des nombres entiers" programme.

Mode of valuation

Continuous assessment in the form of two knowledge tests (60 minutes)

NB: The corresponding credits for this teaching cannot be validated for FGSE students (autumn and spring).

Students from the HEC faculty who wish to follow a Science course at the carré must register with their faculty and inform the secretariat of the Collége des Sciences by email at the following address: cecile.roy@unil.ch

PLEASE NOTE: Information sent by email does not constitute registration!