Non-linear systems are all-around "along the innovation chain" in science, technology, economy, e.g. in areas seemingly so different as cryptography, robotics, gaming, computer-aided design, automated reasoning, thermo-dynamics, logistics etc.
Handling non-linear systems is significantly more difficult than handling linear systems. Usually, non-linear systems are solved by replacing these systems by linear approximations and strategies how to come closer to a solution of the original non-linear systems by iterating the linear approximation. However, by linearization, much information about the non-linear systems is lost. A general and complete direct solution of non-linear systems was a long-standing open problem in mathematics. It has been solved, for the first time, by the Gröbner bases methodology invented by the lecturer in his PhD thesis in 1965. In the meantime, the Gröbner bases methodology is a standard technique implemented in basically all recent mathematical software systems like Mathematica, Maple, etc. Also, some of the
algorithmic ideas behind Gröbner bases can be used as a general problem solving technique in many other areas of computer science.
In this course, we will present the mathematical foundation of the Gröbner bases method and the algorithm for computing Gröbner bases. Then we will show how various hard problems about non-linear systems can be reduced to the computation of Gröbner bases. Many examples of systems of non-linear systems and their solutions in various areas of science, technology, and economy will be used as demo material.
The course will be taught in a style that can be digested by any student who masters algebra at high school level, has good experience in programming in at least one programming language, and has some basic understanding and practice in proving. No special knowledge of mathematics is needed.
Attention:
The course is cancelled this semester. It is planned to hold it next winter term.
Didactic structure:
In the introduction we will give the motivation why non-linear systems are of fundamental importance. Then we will explain the intuitive ideas on which the Gröbner bases approach is grounded. We will then go into the formal details including a complete mathematical proof of the mein algorithm, which can serve for the students as a model how the algorithmic solution of difficult problems can and should be based on rigorous mathematical analysis and reasoning. Many examples and example computations will accompany the presentation so that, at the end of the lecturing part of the course, the students will have a little Laboratory (written in Mathematica) at their hands, which they can use both for applying the method in examples but also for extending the method and playing with improved implementations. This will be the content of exercises. Finally, the students should present the results of the exercises in a short talk (20 minutes) accompanied by an oral exam (30 minutes).
Distribution of ECTS:
* Lecture 15h
* Lecture introduction 0.5h
* Solving the exercises 10h
* Preparing the presentation 20h
* Presentation of exercises solutions and talks 9h
* Preparation for exam 20h
* Oral exam 0.5h
