Some recent trends from research mathematics and their connections to teaching: Case studies inspired by parallel developments in science and technology

We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed relatively constant over a long period of time, they must be varied in accordance with the needs and the demands of students, and they must constantly updated keeping an eye to current research and to modern international trends in technology. Our thoughts and suggestions on the use of these trends in teaching have been tried out by the author, and they are now in textbooks, some by the author.