The aim of the Sparkling Science research project EMMA is to work in close cooperation with students to develop aspects of mathematical algorithmic by means of two ongoing research projects with focus on algorithmic questions. The focus of this cooperation will be on computer experimentations, which are of great importance for research in the field of algorithms. Only by experimenting can the mathematical-algorithmic theory be proven to be right or new approaches be found. Figuratively speaking computer experiments for an algorithmic mathematician equal the experimental research of natural scientists in the laboratory. The experimentations are carried out in large part by the students, thus forming an important bridge between the students and the involved researchers. This will enable the students to directly participate in the research activities and to provide a valuable contribution. The experiments will allow the researchers to convey important aspects of mathematical algorithmic to the students so that a productive exchange can be established for both sides via the experimentations.
In the first research project numerical solution schemes for variational inequalities are developed and analyzed, where mixed finite elements are applied for the discretization. The students are asked to implement and analyse different algorithms for solving the arising systems independently as far as possible and, in a second step, compare them with respect to their convergence properties, dependency of different parameters and efficiency. The second research project is from the field of discrete mathematics and addresses elliptic curves with high rank and Diophantine tuples. In collaboration with the students new rank records are to be established and high rank elliptic curves with prescribed torsion subgroup are constructed, experiments concerning Diophantine tuples are also to be conducted. Applications of the research projects can be found for example in crash simulations and data encryption.
Cooperation between researches and students
The cooperation with the students at the affiliated school HTL Braunau is organized on the basis of the optional course “University Mathematics”, which provides the theoretical basic knowledge and will help establish a concrete framework for the cooperation with the students. As the project progresses, the students will work more and more in small groups in which the algorithms will be implemented and analysed. The performance of experiments, parameter studies and visualization will be supervised by the involved researchers. Subsequently, the results will be presented to the other students in the optional course. Another important part of the cooperation is the preparation of diploma theses in the field of the research projects. To ensure an intense cooperation between the students and the researchers, further activities will be provided in addition to the optional course “University Mathematics”. Presentations at the Salzburger Mathematischen Kolloquium of the deparment of mathematics, several-day workshops outside of the school and university, the chance to visit numerical mathematics and discrete mathematics lectures for the first time and a final presentation at the math.space in Vienna are therefore planed. Teacher trainees of mathematics will also be included in the projects via the university course computational applications. As the school of education is involved in the EMMA project as well, questions concerning teaching methodology will be examined in addition to the research projects, such as to which extent certain principles and concepts are effective. The obtained knowledge will directly influence the further planning and structuring of the optional course “University Mathematics”. During the course of the project it is also planned to create an exemplary guideline of how to include mathematical algorithms in school teaching.