Duration
30h Th, 20h Pr
Number of credits
| Master in mathematics (120 ECTS) | 8 crédits | |||
| Master in mathematics (60 ECTS) | 8 crédits |
Lecturer
Coordinator
Language(s) of instruction
French language
Organisation and examination
Teaching in the second semester
Schedule
Units courses prerequisite and corequisite
Prerequisite or corequisite units are presented within each program
Learning unit contents
In this lecture we are concerned with the following topics in discrete mathematics: finite fields, introduction to cryptography, error-correcting codes, linear recurrent sequences, formal series, p-adic numbers, Ramsey theory, ...
To enlighten this lecture, implementation of the various concepts is given through the use of Mathematica.
This lecture is mainly focused on theoretical aspects, applications are only sketched.
Learning outcomes of the learning unit
The student will master fundamental notions seen during the lectures as well as the corresponding proofs. He will be able to present them clearly and succinctly. Also, he will be able to apply those notions in order to solve related problems.
Prerequisite knowledge and skills
We assume a good knowledge of the concepts of groups, rings, fields and vector spaces.
Planned learning activities and teaching methods
Theoretical lectures using "blackboard and chalk" or beamer, interacting with students. During exercises sessions, students are facing exercises that must be solved and situations that must be modeled on a computer.
Mode of delivery (face-to-face ; distance-learning)
Lectures are mainly dedicated to theoretical aspects. Pratical sessions are devoted to solve exercises and to enlighten the concepts presented during the lecture. It could be considered to implement some cryptographic notions in a computational software like Mathematica. Detailed schedule will be given at the beginning of the academic year.
Recommended or required readings
Lecture notes are available (in french) and can be downloaded from http://www.discmath.ulg.ac.be/
Some complementary material:
- R. Diestel, Graph Theory, 3rd Edition, Graduate Text in Math. 173, Springer, (2005).
- C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Text in Math. 207, Springer (2001).
- E. Seneta, Non-Negative Matrices, An Introduction to Theory and Applications, George Allen and Unwin Ltd, London, (1973).
- J. Buchmann, Introduction to cryptography, Second edition, Undergraduate Texts in Mathematics, Springer, (2002).
- R.L. Graham, D.E. Knuth, O. Patashnik, Concrete Mathematics, A foundation for computer science, Second edition, Addison Wesley, (1994).
- A. Salomaa, Public-Key Cryptography, Second edition, Texts in Theoretical Computer Science, An EATCS Series, Springer, (1996).
- R. P. Stanley, Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics 49 (2012).
- H. Wilf, Generatingfunctionology, Academic Press (1994).
Assessment methods and criteria
The final examination is an oral one : presentation of a selected subject, statements and proofs of theorems and discussion but also direct applications of the theory. In particular, the student will be asked to solve some exercises. Details about the examination will be precisely stated during the year.
Work placement(s)
Organizational remarks
Some useful informations are given on http://www.discmath.ulg.ac.be/ In particular, one has access to the log of the year and also the ones of previous years. This course is organized on academic years starting on an even year (for instance 2018-2019).
Contacts
É. Charlier
Institute of Mathematics (B37) - Allée de la découverte 12 - Sart Tilman, 4000 Liège
Tél. : (04) 366.93.84 - E-mail : echarlier@uliege.be
M. Rigo
Institute of Mathematics (B37) - Allée de la découverte 12 - Sart Tilman, 4000 Liège
Tél. : (04) 366.94.87 - E-mail : M.Rigo@uliege.be